Evolution equations and approximations

- 0 0 for all x € X. The set ~Deff(ip) = {x € X I (p(x) < 00} is called the effective domain of

.

1.6. Convex functionals

and

subdifferentials

41

b)

for all x G X

resp. if and only if (p(x) < liminf ip(xn) for all x G -X" and all sequences (:rn)„6N with w-limn_»oo x„ = x. c) 99 is convex on X if and only if ip((l - X)x1 + Xx2) < (1 - \) 0 there exists a 5 > 0 such that \x - y\ < 6, y G X, implies ip{y) > ip{x) — e. Since we are not interested in the convex functionals ip = —00 or ip = 00, we shall usually assume that the convex functionals under consideration are proper. It is easy to see that a convex functional

c G K,

are closed. b) Assume that

x. This implies tp(x) < liminfn^oo x. We set d = liminfn^oo (p(xn) and choose a subsequence (xnk)keN such that lhrifc.+oo ip(xnk) = d. For any e > 0 we have 0, i.e.,
which proves that

42

Chapter 1. Dissipative

and Maximal Monotone

Operators

the level sets Dc, c e R, are convex and closed or, equivalently, convex and weakly closed (see for instance [La, Theorem 9.2.2, p. 243]). Let (xn)neN be a sequence with w-limn-j.oo xn — x and set d = liminf„^.oo tp{xn). As under part a) of this proof we see that there exists a subsequence (a^jJfceN such that, for any e > 0, we have xnk G Dd+e for k sufficiently large. Since Dd+e are weakly closed, we get x £ Dd+e for all e > 0, so that ip(x) < d — liminf

p(xn). D

1.43. Theorem. Assume that X is reflexive and that ip is a proper, convex and lower semi-continuous functional on X satisfying (1.61)

lim ip(x) = oo.

Then ip has a minimum on X, i.e., there exists an XQ 6 X such that ip(x0) = inf tp(x). Proof. Set rj = inf x e x *p{x) and choose a minimizing sequence (xn)nef$ so that lim„^.oo f(xn) = V- Since

k—*oo

i.e.,
D

The next lemma provides a fundamental property of lower semi-continuous convex functional. 1.44. Lemma. Let (p be a proper, convex and lower semi-continuous functional on X. Then ip is bounded below by an affine functional, i.e., there exist an x* G X* and a constant c G R such that (1.62)

ip{x) > (x, x*) + c for all

x€X.

Proof. Choose XQ £ X and then 0 G R such that (3 < (5 for all x G V{XQ). It is easy to see that epi(y) is a convex subset of X x R. Since

) ^ 0. Obviously V(xo) x (—00,/?) is a convex subset of X x R , which has interior points. Moreover, epi(
1.6. Convex functionais

and

subdifferentials

43

is a separating hyperplane for epi() and V{x$) x (-co, /?). Since (x0, x^)+r -» —oo a s r - > —oo, we must have (x, XQ) +r > c for all (x, r) £ epi(y>). In particular we have if(x)

> C - (X, XQ),

X £

X,

which proves the result with x* = —XQ.

O

For the next result we need the following concept: 1.45. Definition. Let M be a subset of a Banach space X. Then the relative interior intrei M of M is the interior of M in the minimal closed affine subset Y of X containing M. 1.46. Proposition. Let

• (—oo, oo] be a proper, convex and lower semi-continuous functional. Then the restriction Lp |Deft(cz>) *s continuous on intreiDeff( 0 we have to show that N = {x \ \ —e} D {x | tp(x) < e} and the fact that {x | tp(x) > —e} = X \{x \ ip(x) < —e} is open (which follows from lower semi-continuity of
< e}.

The set R is convex and balanced (i.e., x £ R and \a\ < 1 imply ax G R). Moreover, R is also absorbing, i.e., for every x £ X there exists an a > 0 such that ax € R. The latter fact follows from the fact that the scalar convex function g(t) = ip(tx) is finite in a neighborhood of 0 and thus also continuous at 0. The properties of R imply that it is a neighborhood of 0 (see for instance [Wi, p. 32]) satisfying R c N\. For the convenience of the reader we give the short proof that g is continuous at 0. For sufficiently small e > 0 the function g is defined, real valued and convex on U = (—e, e). Let (i„)„gN be a sequence in U with tn ^ 0, n = 1,2,..., and linin^oo tn = 0. From 0 £ (tn,e) for tn < 0 and 0 £ (-e,tn) for tn > 0 we get g(0) < Xng(tn) + (1 - Xn)g(e) resp.

44

Chapter 1. Dissipative and Maximal Monotone Operators

g{0) < Kg{tn) + (1 - A„)c?(-e), where (A„)„ eN C (0,1) with lim„^ 0 0 An = 1. This implies g(0) < liminf #(*„). n—>oo

Using tn G (0, e) for tn > 0 and tn G (—e, 0) for tn < 0 we see analogously that limsup9(i„) < g(0). n—>oo

This proves l i m ^ o o #(*„) = 5(0).

D

1.47. Definition. Let ip be a proper, convex functional on X. a) The subdifferential dip(x) of 9? at x G X is the set dip(x) = {x* € X* I ¥>(y) - ip(x) > (y- x, x*) for all y G X}. b) The functional tp* : X* ->• (-00,00] defined by V?*(p) = sup((x,p)-
peX*,

is called the conjugate of (^ or the Fenchel transform of ip. If y> = 00 or ip = —00 then we would have dip(x) — X* for all x G X, a case which is of no interest to us. 1.48. Theorem. The conjugate functional ip* of a proper, convex and lower semi-continuous functional ip is again proper, convex and lower semi-continuous. Proof. Convexity of (p* is an easy consequence of the definition of ip*. Since

—00 for all p G X*. In order to show that ip* ^ 00 we take p = x*, where a;* is an element in X* satisfying (1.62). Then we have {x,p) — tp(x) < —c for all x G X, i.e., • p. For e > 0 we choose x€ G X with {x€,p)-ip(x€) >ip*{p)-e. Then (x e ,p„) - ^(z,:) < <£*(?«) for all n = 1,2,... implies liminf ip*(pn) > lim ((xf,pn)

- ip(xe))

= (z e ,p) - ip(xt) > ip*(p) - e, which proves ip*(p) < liminfn^oo ip*(pn).

D

If ip is a proper, convex functional on X, then dip(x) = 0 for any x with <£>(#) = 00. Therefore we have (1.63)

domdip cDef[(
1.6. Convex Junctionals

and subdifferentials

45

for the operator x -> dtp(x). That in general we do not have equality in (1.63) is shown by the functional ip(x) — — \x\x/2 for x < 0 and 0. For this functional we have Deff(y) \ domdy — {0}. 1.49. Proposition. Let ip be a proper convex functional on X. Then the operator dtp C X x X* is monotone. If, in addition,

0

for all Xi G domdtp, x* G dip(xi), i = 1,2, with x\ /

x2.

Proof. Let x* G dtp(xi) with Xi G domdtp, i = 1,2. Then we have ip(xi)-tp{x2)>{xi-x2,x*2)

and

y ^ ) - ip{xi) > (x2 - x1,x*1)

which implies 0 < (x± — x2, x\ — x2). Assume now that

x0,xl).

We set y = (x0 + y0)/2 and get (zo))/2 = ((y0 X0)/2,XQ) = (y — xo,x%), a contradiction to XQ G dip(xo)- Therefore we have {y - x, x*)

for x, y G X with x ^ y and x* G dip(x).

Then strict monotonicity of dtp follows analogously as above monotonicity.

•

1.50. Example. Let tp be a proper convex functional on X which is Gateaux differentiate at some x G domcV, i.e., there exists a w* G X* such that (1.64)

l i m

t|0

^ +

fa)-^)=

i o r a l l v £ X

.

t

Of course, w* is uniquely determined and is called the Gateaux differential of ip at x, w* =
^ + ^ 7 ) ) - ^ ) < ^ ) - ^ ) , o

(y-x^'(x)},

which proves ip'(x) G dtp(x). Conversely choose w* G dip(x). Then we get, for any y G X and t > 0,

46

Chapter 1. Dissipative

and Maximal Monotone

Operators

Again using Gateaux differentiability of ip at x we get, for t \, 0, (y,(y,w*)

for all y G X,

which implies w* =
tp(x) = -\x\2,

xeX.

Then

where F denotes the duality mapping on X. Indeed, choose first x* G Fx. Then we have (y - x, x*) = (y, x*) - \x\2 < \y\ \x\ - \x\2

<\\y\2-\\x\2,

v ex,

which proves x* G d
+ ty\2-\x\2)<\x\\y\

+ ^\y\2,

y G X, t > 0.

This, for 11 0, implies (y,x*) < \x\ \y\, y & X, i.e., we have |x*| < |x|. On the other hand we get (-2t + t2)\x\2 = \x - tx\2 - \x\2 > 2t(-x,x*), t > 0, resp. 2(x,x*) > (2 - i)|a;|2, t > 0, which for 11 0 implies |a;|2 < (x,x*) < \x\ \x*\, i.e., we have also \x\ < \x*\. Thus we have shown |x*| = |x| and (x,x*) = |x| 2 i.e., x* G Fx. 1.52. Example. Let K be a closed convex set in X and set

, .

IK(X)

(1.65)

dIK(x)

= {x* G X* | (x - t/,x*) > 0 for all y e A"}.

We have dom9/K — K = Defr(/K')- From (1.65) we conclude d/*r(z) = {0}

for x £ K, o

which coincides with the fact that 0 there exists a y\ such that lim^o \y\\ = 0 and x + Ay + Ay* G # .

1.6. Convex functionals

and subdifferentials

47

Then for any x* G (9ix(x) we have (v + V\,x*)<0,

A>0,

which for A 4.0 implies (y,x*)
Note that for x G K the set of all tangential directions is X, so that x* = 0. Since K is convex, for any z G K the direction y = z — x is a tangential direction at x. Therefore we can also characterize dlx(x) as dIK{x)

= {x* G X* \(y,x*) < 0 for all tangential directions for AT at a;}.

In geometrical terms this means that at a point x G dK (= boundary of K) the subdifferential 8IK(X) is the cone of outer normals to K at x. The main theorem of this section is: 1.53. Theorem (Rockafellar). Let if be a proper, convex and lower semicontinuous functional on X. Then dip is a maximal monotone operator. Proof. We shall give the proof under the additional assumptions that X is reflexive and that X and X* are strictly convex. For the general case we refer to [Rol]. By Theorem 1.37 it suffices to prove that range(i ? + dip) = X*. We fix XQ G X* and define the functional / by f(x) = -\x\2+


xeX.

It is easy to see that / is proper, convex and lower semi-continuous. Moreover, by Lemma 1.44 there exist an x* G X* and a c G K with ip(x) > {x, x*) + c for all x G X. This implies lim^i^oo f(x) = 00. By Theorem 1.43 there exists an XQ G Deff (/) = Deff() with f(x0) = inf xeX f{x). This implies f(x)-f(x0) >0 for all x G X or, equivalently, (1 66)

^

~ ^X^

" ^ ~ X °' X o) + 21 210 ' 2 ~ 2 ^ > (x — x0,XQ) — (x — xo,Fx),

x G X,

where we have also used the fact that F is the subdifferential of x —» \x\2/2 (see Example 1.51). For arbitrary u G X let xt = x0 + t(u — Xo), 0 < t < 1. Then we get from (1.66) and convexity of ip p{u)

-P(XQ)

> -(p(xt)-p(x0)) = (u -

XQ, XQ)

> -(xt-xo,x*0) - (u - X 0 ,

Fxt).

- -{xt

-x0,Fxt)

Chapter 1. Dissipative and Maximal Monotone

48

Operators

Observing that F is demi-continuous (see Proposition 1.1, b)) we get, for 11 0, the inequality (u — XQ,XQ — FXQ) for all u £ X, which proves XQ - Fx0 G dip(x0), i.e., XQ G {F + dip)x0. • 1.54. Proposition. Let

0 there exists a (3 > 0 such that [u,v] G dtp and \v\ < a imply \u\ < /3 (i.e. dom(dip)-1 = X* and (dtp)-1 maps bounded subsets of X* onto bounded subsets of X). Proof. Assume first that ip is radially unbounded and let [x„,x*] G dip, n = 1,2,..., be a sequence with limn-^oo |x„| = oo. By definition of dip we have (y — xn, x*) for all y £ X. We choose a y0 G Deg(
i

ip(xn) -^

i

i

ip(y0) i

ii

n

—

l,

z,....

which implies rim„_>.oo(^n - yo,x*n)/\xn\ = oo. Assume now that dip is coercive and choose a > 0. Since dip is maximal monotone (see Theorem 1.53) we get from Theorem 1.39, b), that range dip = X*. Assume that there exist [u n ;^n] G dip such that \vn\ < a for all n and linin^oo \un\ = oo. Then the estimate (un - x0,vn) i—j \un\

\un - x 0 | < —;—:—a —> a \un\

as n -4 oo

gives a contradiction to (1.46). Therefore (iii) holds. Finally we assume that (iii) is true. Let (j/n)neN C X be a sequence with lim„_4.oo \yn\ = oo- For fixed a > 0 we define the elements vn = a|j/„| _ 1 Fyn, n = 1,2,..., where F is the duality map X -> X*. Then for any n there exists a un G X with [un,vn] G dip. Since \vn\ — a, we get |u n | < (3, n — 1,2,... . For j/o G X we have ip(y0) - ip(un) > (y0 - un, vn), which implies ip(un) < ip(y0) + a(3+ \yo\a, n = 1,2,... . This and Lemma 1.44 together with |u„| < (3, n = 1,2,..., prove that the sequence (ip(un)) N is bounded. From the definition of dip we get tp(yn) - ip(un) > {yn - un, vn) = a\yn\ - (un, vn) > <x\Vn\ ~ a0, which implies jg(jfa) ^ ip(u) - a/3 + |j/n|

n =

1 2

|j/n|

For n -> oo we obtain liminfn_+00 v?(y n )/|j/ n | > a. Since a was arbitrary, we get the result. •

1.6. Convex junctionals

and

subdifferentials

49

1.55. Theorem. Assume that X is a real Hilbert space with inner product (•, •) and that A is a maximal monotone operator on X. Let if be a proper, convex and lower semi-continuous functional on X satisfying dom ACidomdip ^ 0 and (1.67)


< ip(x) + AM,

A > 0, x e D eff (p),

where M is some non-negative constant. Then the operator A + dip is maximal monotone and6 \A°x\ < \(A + dip)°x\ + M1/2,

Z € dom,4 nDeff()•

Proof. According to Theorem 1.53 the operator dp is maximal monotone on X, so that —A and — dip are m-dissipative. By Lemma 1.19 the minimal section A0 of A is single-valued and dom A 0 = dom A (note that dom A0 = dom(-A) 0 and (—A)°x = — A°x, x £ dom A 0 ). According to Lemma 1.23 there exists, for any A > 0 and any y £ X, an element xx £ domdip such that y£xx-

(-A)xxx

+ dip{xx).

Moreover, \xx\ is bounded on A > 0 by Theorem 1.24, a). Observing y — xx + {-A}xXx £ dip(xx) we obtain, for any z £ X, (1.68)

ip{z) - ip(xx) >(z-xx,y-xx l

We take z — (I + \A)~ Xx, (1.68) \({-A)xxx,

y-xx

+

(~A)xxx).

so that z — xx = A(—A)xXx-

+ {-A)xxx)

< ip{{I + XA^xx)

Then we get from

- ip(xx) < AM,

A > 0,

and 1(-A)xxx\2

< \(-A)xxx\

\y - xx\ + M.

Since \xx\ is bounded on A > 0, the last estimate shows that |(—A)xXx\ is also bounded on A > 0. According to Theorem 1.24 the operator —A - dip is m-dissipative or, equivalently, A + dip is maximal monotone. For x € Deff(
= ( ( / + XA^x < AM,

i.e., ((-A)xx,p) (-A)°x = -A°x, (1.69)

< M. so that

- x,p) <
-
A > 0,

By Theorem 1.20, b), we have \imx\.a(-A)xx

(A°x,p) > - M

=

for all p £ dip{x).

If we set y — (A + dip)°x, then y = y\ + yi with y1 £ Ax and y-2 £ dip(x). Let t/i = A°x + u. Then {A°x, u) > 0, because |^4°ar| = inf ze ^ x \z\ and Ax is 6

Recall that A0 denotes the minimal section of A (see Definition 1.18).

50

Chapter 1. Dissipative

and Maximal Monotone

convex (see Lemma 1.14). Therefore we have (A°x,yi) we get

Operators

> |yl0a;|2. Using (1.69)

(A°x, y) = (A°x, yi) + (A°x, y2) > \A°x\2 - M and |,4°:r| 2 < \A°x\\(A + d 0, and h(0) = -y1/2 we get the estimate a < 0 + 7 1 / 2 . D 1.56. Example. Let Q, be a bounded open set in Rd with sufficiently smooth boundary. 7 We define the functional

(—00,00] by

(1.70)


= \jaW<*)\l
I 00

for u e

^o(n),

otherwise.

1.57. Lemma. The functional ip defined by (1.70) is proper, convex and lower semi-continuous. Proof. It is obvious that ip is proper and convex. In order to prove that

is also lower semi-continuous we choose u G L 2 (0) and set a = liminf^_>.„ 0. We only need to consider the case a < 00. We choose a sequence (W„)„ 6 N with \u — UU\L2 —• 0 and 00. By definition of

00((^, un)Hi = (4>,v)Hi for all <j> G HQ(Q). Since (u„)„ e N i s bounded in HQ(Q) and HQ(£1) as a Hilbert space is uniformly convex, we get lim \v — un\Hi = 0. n—foo

° 2

By the continuous embedding HQ(Q.) Q L (Q) this implies l i m ^ o o \v — un\i,2 = 0, so that we have v = u. From p(u) - ip(un) = ((u,u)Hi - {un,un)Hi)/2 = ((u,u- un)Hi + (uun,u„)Hi)/2 < \u - u n | H i ( | u | H i + M)/2 -> 0 as n -> 00 we see that a — limn^oo ip(un) =
1.6. Convex functionals

and subdifferentials

51

1.58. Lemma. The duality mapping F : -ffo(fi) -> # _ 1 ( 0 ) is given by Fu = -Au1

uG^(fi),

where derivatives are taken in the distributional sense. Proof. By Proposition 1.1, c), the duality mapping F is single-valued and continuous. In view of the dense and continuous embeddings 8

Hl(p) Q L2{Q) Q H-\fy the bounded linear functionals on H$ (Q) are represented by the L 2 -inner product: For / G L 2 (fi) we have {u, f) = (u,f) for all u G HQ(Q) and, for arbitrary (u,f) = lim (u,fn) = lim (u,fn),

u e #o(fi),

where (/ n ) n eN C L2{Q) is any sequence with | / - fn\H-i -» 0. From Lemma 1.3, c), we obtain, for f,g G Co°(fi), / / 0, 1

-<<7^/> = (5,/)+ = (9J)-

= il\f

= ^ ( _ / IV/ + tWg\lddx)

+ t9\m ' |t=Q=

j^—Jjyf(x),Vg(x))Kddx.

This implies by Green's first identity (2,*7> = [(Vg(x),Vf(x)h*dx

= - fg(x)Af(x)dx

for f,g£ Cg°(n). By density of C£°(fi) i n # o ( n ) t h i s Therefore we have Ff = -Af For / e flo(fi) of .F we have

w e choos

(g,Ff)=

8

i s t r u e for a11

e (/«) C C£°(fi) with \f - fn\Hi

lim f

/(V$(ar), V/(a;))H-da:,

i.e., the spaces i?o(fi), L2(Q) and if

9 G #d( f i )-

for / G C0°°(n).

lim (g,Ffn)= -

= -(g, A / )

1

-> 0. By continuity

(Vg{x),Vfn{x))%ddx
(SJ) form a Gelfand triple (see page 90).

52

Chapter 1. Dissipative and Maximal Monotone

Operators

On the other hand we have, for g G Cg°(fi), d

3=1

J

j=l

= iljjtJ{x)ltj{x)dX

=

Jjy9(x),Vf(x))Rddx = {g,Ff).

This proves F / = - A / for / e i ^ ( f i ) .

D 2

2

1.59. Proposition. The subdifferential dip C L (Cl) x L (Q) of the functional ip on L2(Q) defined by (1.70) is given by H2(tl),

domdtp = H^(Q) n d
uGdomdip,

i.e., —d

H^(n)r)H2(n).

From (u, Au) = (u,Fu) = |«|^i > 0, u G dom A, we see that A is monotone as an operator C L2(Q) x L 2 (0). By Corollary 1.40, rangeF = H~1(fl) and consequently range A = L2(Q). We choose u G dom A Then, for any w € HQ(£1), we get (observing also -Au = F u and \Fu\H-i = \u\Hi) 0 '

(w — u, —Au) = (w, Fu) — (u,Fu) < \w\Hi \u\Hi — \u\2Rl

o

1,

,

1,

i.e., Au = -Au G dip(u). Thus we have shown that A c dip. Assume that, for some uo, WQ G L2{Q), we have {U — UQ, AU — WQ) > 0 for all u G dom A Since ranged = L2(il), there exists a i i i G dom A with w0 = At«iTherefore we have (U-UQ, A(u-U\)) = {u—uo, F(u—ui)) > 0 for all u G dom A F o r u = (uo + ui)/2 we get (tii-u 0 ,-F(wo-wi)) > 0. Monotonicity of F implies the converse inequality, i.e., we have {uo — UI,F(UQ - u\)) = |w0 - wil^i = 0.

1.6. Convex junctionals

and subdifferentials

53

Thus we have wo = u\ G domA and WQ = AUQ. Thus A is maximal monotone. Since dip is a monotone extension of A, we have A = dip. O 1.60. E x a m p l e . Let fi C Rd be bounded and assume that j : RN -> (-oo, oo] is a proper, convex and lower semi-continuous functional. We define the functional i>: L 2 (ft; RN) -> (-oo, oo] by Deff(^) = { n e L2(ii;RN)

(i.72) U X

^/ u \ = \JsiJ( ( ))

dx

1 oo

| w(a;) G DeffG0 a.e. on fi and

jM-M^ai)}, D e f f (VO,

fOT U G

otherwise.

According to Lemma 1.44 there exist a vector a G M^ and a constant c G K with j{£) > (a,f)Kjv + c for all f G R w . This implies

V>(w•) >

/ ((a,u(x))Kjv + c) da;

> cmeasfi — |a| R jv(measfi)

1/2,

M£2(Q.ajv) > —oo

for all u G L 2 (fi;K w ). It is easy to see that ip ^ oo and that ij) is convex. In order to prove lower semi-continuity of tp we choose u G L2(Cl; RN) and set a = liminf w _y u ^(t;). Only the case a < oo needs to be considered. We choose a sequence (w„)„eN such that \u — Mn|z,2(n;RJV) ~* 0 a n d ip(un) —>• a. We can also assume that u n (x) —>• u(a;) a.e. on 0. From \u — un\L2^.^N^ —>• 0 we get (1.73) n

lim / ((a, u„(a;))Hjv + c) da; = / ((a, u(a;))Rjv + c) dx. ->°° Ju Jn

Since j( u n(£)) - ((a,un(x))^N (using also (1.73)) a-

+ c) > 0 a.e. on ft, we get by Fatou's lemma

/ ((a, U(X))UN +c) dx = lim inf / (j(un{x)) n Jn ->°° JnK > / (liminf j(un(x)) JoV n—*oo

> / j(u(x)) dx-

Jn

- ((a, un{x))RN + c)) dx ' - lim ((a,un(x))RN n—*-oo

I ((a, U(X))RN

Jn

which implies a > ip(u), i.e., ip is lower semi-continuous.

+ c) ) dx ' /

+ c) dx,

54

Chapter 1. Dissipative and Maximal Monotone Operators

1.61. Lemma. The subdifferential dtp of the functional ip defined by (1.72) is given by dtp = {[u,v] G L2{Q;RN)

x L2{n-,RN)

\u(x) G domdj a.e. on Q, j{u{-)) G L\n)

and

v{x) G dj(u(x))

a.e. on f2}.

Proof. Let first u,v G L2(£l;RN) satisfy the conditions given in the lemma and choose w G L2(Q;RN). Then j(w(x)) — j(u(x)) > (w(x) — u(x))v(x) a.e. on 0 . This implies tp(w) — tp(u) = / j(w(x))dx

(1.74)

-

j(u(x))dx ,/n

^J?

> / (w{x) — u(x)Jv(x) dx = (w — u,v), which proves [u, v] G dtp. Conversely let [u, v] G dtp be given. Since we have &om dip C Deff (xp), we see that u(x) G Deff(j) a.e. on ft and j(u(-)) G L x (fi). For arbitrary iu0 G RN and any measurable set Z C 0 we define . I w0 una;) v ; = <

for x G Z,

[u(x)

Since (1.74) is true for any w G L2(Q;RN),

foix£tt\Z. we conclude that

/.<{j(wo) - j(u(x)) - (w0 - w(a;))t;(a;)) dx > 0. >z This implies that for any w G M^ there exists a set Z(w) of measure zero such that j(w) - j(u(x)) >(w-

u(x))v(x)

for all x G ft \ (£(«;) U Z0),

where ZQ is a set of measure zero with u{x) G Deff (j) for all x G H \ ZQ. It is clear that Z(w) = 0 for w <£ Deff(j). Let (w„)ragF} be a sequence comprising all rational vectors in int re iD e ff(j). We set Z = Z0 U U^Li Z{wn). Then measZ = 0 and j{wn) - j{u{x)) > (wn — u(x))v(x) for all n = 1,2,... and all x G fi \ Z. By continuity of j on intrei Deff(j) (see Proposition 1.46) this implies (1.75)

j(w) - j(u(x)) > (w -

u(x))v(x)

for all w G int re i Deff(j) and all x G 0. \ Z. For u>0 G D eff (j) \ int re iD e ff(j) we choose w\ G intrei Deff(j). Then \WQ + (1 - X)wi G int re i Defr(j) for 0 < A < 1 and limsup A | 1 J(\WQ + (1 — A)wi) < j(wo). By lower semi-continuity of j we have liminfAti j(Au;o + (1 — A)u?i) > j(wo). Thus we have limA-fi j(\wo + (1 — A)u>i) = i(^o). Then (1.75) implies j(wo) - j{u{x)) > (wo - u(x))v{x) for all

1.6. Convex junctionals

and

subdifferentials

55

wo e Deff(j) \ int re iD e ff(j) and all x € fl \ Z. Since (1.75) is trivially true for all w £ DeffO), we have shown that it is true for all w G RN. This implies u(x) € domdj and v(x) € dj(u(x)) a.e. on fi. D 1.62. Example. Let Q be a bounded open subset of Md with sufficiently smooth boundary and set A = dip, where dip is given by (1.71). Furthermore, we set B = dip, where dip is given by Lemma 1.61 for AT = 1. The operator A + B = dip + dip = —A + dip is a semilinear elliptic operator. As an application of Corollary 1.25 we have: 1.63. Proposition. The operator A + B = dip + dip is maximal monotone on L2(Q). In particular we have d(ip + tp) = dip + dtp. Moreover, [u,v] e d{

uei/01(«)nff2(n), u(x) € domdj a.e. on 0,, j{u(-)) e Ll(tt) and v(x) + Au(x) € dj(u(x)) a.e. on fl. Proof. Since the operators A and B are maximal monotone, the operators —A and — B are m-dissipative. We have to investigate (—Au, (—B)\u) for u e d o m A We fix u e dom^l and set v = (I + XB)~1u e d o m £ . From Lemma 1.61 we see that v(x) = ( / + Xdj)~1(u(x)) a.e. on $X This and = A- 1 ((/ + XB^u

{-B)xu

- u)

imply ((~B)xu)(x)

= (-dj)x(u(x))

a.e. on ft.

From Theorem 1.10, (i), we get, for T\ < r 2 and A > 0, {-djhfo)

- (-dj)x(n)

= ! ( ( / + Xdj)~lT2 - (I + Xdj)-1^ < -r

(T2

- ri - |r 2 -

TI|)

- (r 2 - n ) )

= 0,

i.e., (-dj)x is decreasing on M. Moreover, by Theorem 1.10, (vi), (-dj)x is globally Lipschitzian. Therefore (-dj)'x exists a.e. on K and (—dj)'x < 0 a.e. For u € domA = H$(n) D # 2 (ft) we get, for A > 0, {-Au,{-B)xu)

=

Ja

iAu{x){-dj)x{u(x))dx

= ~ J{-dj)'x{u{x))\Vu{x)\lddx

> 0.

By Corollary 1.25 we conclude that —A — B is m-dissipative, i.e., A + B = dip + dip is maximal monotone on L2(Q). This implies also d(ip + ip) = dip + dip,

56

Chapter 1. Dissipative

and Maximal Monotone

Operators

because d
CHAPTER 2

Linear Semigroups In this chapter we present some of the essential facts on linear Co-semigroups and linear abstract Cauchy problems. The main purpose of this chapter is to provide basic notions in the conceptually much simpler linear context which will later be generalized to nonlinear situations. The core of this chapter consists of a thorough investigation of the connection between Co-semigroups and mild resp. strong solutions of linear abstract Cauchy problems including Ball's theorem (Section 2.2) as well as of complete proofs for the Hille—Yosida theorem (Section 2.3) and the Lumer-Phillips theorem (Section 2.4). In Section 2.5 we present an application of the Lumer-Phillips theorem to a variational formulation of a second order equation. Concerning the general theory of linear Co-semigroups we refer to [Hi-Ph], [Pal] and [Go]. Naturally, many important aspects of the theory for linear Co-semigroups are neglected in our presentation. In addition to the books already quoted above we refer to [C-H-A-D-P] for adjoint semigroups, to [Fa] for cosine operators and to [DP-Gr] for maximal regularity results. In this chapter X denotes a real or complex Banach space, when not otherwise stated.

2.1.

Examples and basic definitions

We start with some simple and well-known examples in order to motivate some of the fundamental definitions which we introduce in the following and to illustrate also the wide range of applicability of the abstract theory. 2.1. Example. Consider the following initial value problem for a first order partial differential equation on (0,1): (2 1)

glw(t,x)

+ -^w(t,x)=0,

0<x0,

2

w(0,x)=
58

Chapter 2. Linear

Semigroups

We define the linear operators S(t), t > 0, on X = £ 2 (0,1) by

Note that S(t)(j) = 0 for t > 1. The following properties of the operators S(-) are obvious: (a) \\S(t)\\ < 1,

£ > 0 (boundedness).

(b) 5(0) = / and S(t)S(s) = S(t + s), t,s > 0 (semigroup property). (c) \S(t)(f> — \x —> 0 as t 4- 0 for all <j> G X (strong continuity at t = 0). Note that (c) is equivalent to / \4>{x -t)-

(x)\2dx + / \4>{x)\2dx -> 0

as 110.

For (/. G -fiT^O.1) with <£(0) = 0 we see that also S(t)<j) G ^ ( 0 , 1 ) and (5(t)^)(0) = 0. Moreover, we have {d/dt)(S{t)){-) = ~(d/dx)(S{t)4>){-) for all t > 0. This shows that the function w(t,x) — (S(t)4>)(x) satisfies (2.1) in L 2 (0,1) for t > 0 . We define the linear operator A on X by dom A ={4>€ Hx(0,1) | (0) = 0}, A(j) = -(j)',

4>£domA.

Then, for any (j> G dom A, the function u(t) = S(t), t > 0, satisfies M(-) G C ( 0 , o o ; d o m A ) n C 1 ( 0 , o o ; X ) , where |<^|domJ4 = |'|x (note that this norm is equivalent to the graph norm \4>\x + \'\x on dom A). This follows simply from the fact that, for G dom A, the function : (-co, 1] -*• R defined by

kr)=l° V ;

forT<0

'

|<£(r) for 0 < T < 1, is in Hl(—co, 1) and S'(i) is just the left-shift. Moreover we see that, for <j> £ dom A,

(2.3)

! « ( * ) = Atf),

*><>,

u(0) = <j>. Thus we have seen that for smooth initial data, namely <j> G dom A, the partial differential equation (2.1) is equivalent to the abstract Cauchy problem (2.3).

2.1. Examples and basic definitions

59

Since dom^l is dense in X, for any <> / G X, there exists a sequence ((j>n)neN with \4> — 4>n\x —> 0 as n -> oo. According to (a) from above we have lim S{t) uniformly for t > 0. In this sense u(t) = S(t) can be called a generalized solution of (2.3) resp. w(t,x) = (S(t)(j))(x) a generalized solution of (2.1). 2.2. Example. In this example we consider the one-dimensional heat equation which governs the temperature distribution T(t, x) in a thin homogeneous rod of length one, jjjT(t, x) = ^T(t, 2 4

(-)

x),

T(t,0) = T ( * , l ) = 0 ,

t>Q,0<x0,

T(0, x) = 4>{x), 0 < x < 1, <j> G L 2 (0,1). Here we assume that the quotient of mass density and heat conductivity is normalized to one. If (2.4) has a solution T(t, x) which is in C 2 (0,1; K) then it is given by T(t,x) — (S(t)4>){x), where the solution operators S(t), t > 0, are given by oo

(2.5)

Y,e-{k7l)2tfaek,

S(t)<j> = fc=i

where the orthonormal basis (ek)k=i,2,... of X = L2(0,1) is defined by ek(x) = V2smkTrx,

0 < x < 1, A; = 1,2,... .

Moreover, , 0fc = (,efc)x = /
A; = 1,2,...,

oo

0 = ^ ^ e * in X. fc=i

Note that the definition in (2.5) makes sense for all G X. It is easy to see that the operators S(t), t > 0, satisfy the properties (a) - (c) listed under Example 2.1. For the present example we define the operator A by dom A = H%(0,1) n # 2 ( 0 , 1 ) , OO

A(f> = " = - ^2(kir)2(f>kek

OO

for <j> = ^

0 fc e fc G X ,

fc=i fc=i 1

where Ho (0,1) = {<j> G ff^O.l) | 0(0) = 0(1) = 0} and H2(0,1) = {<j> = XlfcLi^fcefc I Efcli(fc7r)2(^fcefe e x } - Using .Aefc = -(kir)2ek we see that, for

60

Chapter 2. Linear

Semigroups

G dom A, 1-S(t)cf> = AS(t), at

i>0,

i.e., the function u{t) = S(t)(f>, t > 0, is a solution of —u(t)=Au(t),

t>0,

u(0) = 4>. Let G X. Then, for any ^ = Y^k=\ ^fcefc>the function t -» (V>, S{t)4>)x = Yl'kLi e - ^ *"^k4>k is analytic on (0, oo), because the series for (ip, S{t)<j>)x and its derivatives are uniformly convergent on any interval [6, oo) with S > 0. Thus t —> S 1 ^)^ is weakly analytic on (0, oo) and therefore also strongly analytic (i.e., as a function (0,co) -» X). Moreover, using the estimate 7 2/ e -2n*

< (£/e)M J - ,

t

> o, s > 0, I = 1,2,...,

we can prove that, for any <j> G X, we have S(t)<j> G d o m ^ , I = 1,2,..., and 1^5(4)^ < i ^ l | 0 | x ,

t>0, £=l)2)... .

The examples given above motivate the following definitions: 2.3. Definition, a) Let S(t), t > 0, be a family of bounded linear operators X —> X. The family S(-) is called a Co-semigroup on X (also strongly continuous semigroup) if and only if the following properties hold: 5(0) = I, S(t)S(s)

= S(t + s),

t, s > 0,

(semigroup property)

lim \S(t)x — x\x = 0 for all x G X (strong continuity at 0). b) Let £(•) be a Co-semigroup on X. The linear operator A defined by dom A = \x G X | lim-(S(t)x Ax = lim-(S(£)a; — x),

— x) exists},

x G dom A,

is called the infinitesimal generator of S(-). The following proposition shows that we always have an exponential bound for a Co-semigroup.

2.1. Examples and basic definitions

61

2.4. Proposition. Let S(-) be a Co-semigroup on X. stants M > 1 and w £ R such that ||S(t)|| < M e u t ,

(2.6)

Then there exist con-

i>0.

Proof. We first prove that there exists a to > 0 such that ||S(i)|| is bounded on [0, to]- Assume this is not true. Then there exists a sequence (t„) n€ N of positive reals such that \imn-+ootn = 0 and limn^oo ||5(t„)|| = oo. By the uniform boundedness principle there exists an x € X with linin-^oo \S(tn)x\ = oo, a contradiction to strong continuity of S(-) at t = 0. Thus we can choose M > 1 and to > 0 such that \\S(t)\\<M,

0
For arbitrary t > 0 we choose k G No and T G [0, to) such that t = fct0 + T. Then by the semigroup property we get ||5(t)|| <

\\S(T)\\

\\S(t0)\\k < Mekln^s(-tM

< Mtoewt,

where u = to" 1 In ||5(t 0 )|| and M to = Mmax 0 < T < t o e -(r/to)in||S(to)|| ( = 1

||S(t 0 )|| > 1 and = MH^to)!!" if ||5(t 0 )|| < 1)-

M

if

•

2.1. Remark. By Proposition 2.4 we see that ||5(t)|| is bounded on any compact t-interval. The proof of Proposition 2.4 shows that, for any T > 0, we have ||S(f)|| < MTeUrt, t > 0, where u)T = T~1 In ||5(r)|| and MT > 1. 2.5. Corollary. Let S(-) be a Co-semigroup. Then, for any x £ X, we have S{-)xeC{0,oo;X). Proof. The result follows from S(t + h)x - S(t)x = S(t)(S{h)x h>0, resp. from S(t - h)x - S(t)x = -S(t - h)(S{h)x -x),t>0,h£ together with Proposition 2.4 and strong continuity at t = 0.

-x),t>0, (0,t), D

Proposition 2.4 motivates the following definition: 2.6. Definition. Let S(-) be a Co-semigroup on X. a) The number Q = inf {u> e R |there exists an M > 1 such that \\S(t)\\ < Me"\

t > 0}

is called the (exponential) growth rate of S(-). b) The semigroup S(-) is of class G(M,u)

if and only if \\S(t)\\ < Me"*, t > 0.

Since (2.6) implies limsupt^^ t _ 1 In ||5(t)|[ < w, we see that limsupt _ 1 ln||5(t)|| < Q. t—fOO

62

Chapter 2. Linear

Semigroups

Let u> = inftxji-MnllS^)!!. Then Q < lim n a V ^ r " 1 In ||5(t)||. For e > 0 there exists a te > 0 such that t~l In ||5(t £ )|| < to + e. In view of Remark 2.1 from above there exists an Me > 1 such that ||5(i)||<M£e^+£)*,

t>0.

_1

This implies Q < cD and limt->oo£ In IIS'C*)!! = u>. Thus we have proved the following result: 2.7. Proposition. Let S(-) be a Co-semigroup on X. S(-) is given by

The growth rate w of

Q=limUn\\S(t)\\=}nd\n\\S(t)l t—>-oo t

t>0 t

2.8. Example. Let A be a bounded linear operator on a Banach space X and set 00

(2.7)

fk

S(t) = £ - A f c ,

*^°-

fc=0

Then it is clear that S(-) satisfies properties (b) and (c) listed under Example 2.1. Instead of (a) we have \\S(t)\\ < ell*"*,

i>0.

The family S(-) of operators defined by (2.7) is norm continuous on t > 0, i.e., S(-) G C(0, oo; C(X)), which is not true for the families defined in Examples 2.1 and 2.2. We can also recover the operator A from S(-) by Ax = lim - v(S(t)x -x), t|o t '

x e X.

The next theorems contain some important facts about Co-semigroups: 2.9. Theorem. Let S(-) be a Co-semigroup on X with infinitesimal generator A. Then, for any x e dom A, we have S(-)x 6 C(0, oo; dom A) n C^O, oo; X) and -rS(t)x = AS(t)x = S(t)Ax, t > 0. at Proof. Let x e dom A be given. For t > 0 and h > 0 we have i (5(t + /i)z - S(t)x) = S(t) ( i ( S ( h ) x - a:)) = ~(S(h)S(t)x For ft 4- 0 this implies S'(t)a: G dom A and —S(t)x = S(t)Ax =

AS(t)x.

S(t)x).

2.1. Examples and basic definitions

63

This proves that t -> AS(t)x is continuous, which together with continuity of t ->• 5(t)a; proves S(-) G C(0, cojdom^). For t > 0 and ft G (0, t) we have -Us(t

- h)x - S(t)x) - S(t)Ax\ < S(t - h) (j- (S(h)x - x) < Me^lUsityx

- x) -

S{h)Ax)

S{h)Ax

as h \. 0, where we assume without restriction that u > 0. This proves -rS(t)x at

= S(t)Ax,

t > 0.

a 2.10. Theorem. Let S(-) be a Co-semigroup on X with infinitesimal generator A. a) The operator A is closed and densely defined (i.e., domj4 = X). for any x G dom A and t > 0 we have jQS(s)xds G dom A and (2.8)

Moreover,

A f S{s)x ds = S(t)x - x. Jo

b) For any f G ^ ' ^ ( [ O , o o ) ; ^ ) 1 we have / S(t - s)f(s) ds e dom ^4, (2-9)

t > 0,

J

°

A

t

t

S(t~

S(t- s)f'(s) ds, t > 0. Jo Jo Proof, a) Let (a;n)n6N be a sequence in dom A with xn —>• x and Axn —• y as n —> oo. From Theorem 2.9 we get (2.10)

s)f{s) ds = S(t)f{0) - f{t) +

S'(i)x„ -xn=

[ S(s)Axn ds, t > 0. Jo Using |5(s)^4a; n -5(s)j/| < M\Axn~-y\, 0 < s < t, for some M > 1 we conclude that S(t)x-x=

I Jo

S{s)yds.

Continuity of s —> S(s)y implies lim^(S(t)x-x) lw

= S(0)y = y,

ioc([°.°°); x ) = { / e L ioc([°-°°);*) l t h e r e e x i s t s a s e ^ l o c U 0 . ° ° ) ; * ) w i t h /(*) =

/(0)+/oS(s) t>0.

rfs f o r

t > 0 } . In particular f'(t) exists a.e. on t > 0 and /(*) = / ( 0 ) + / 0 ' / ' ( r ) dr,

64

Chapter 2. Linear

Semigroups

i.e., x £ domA and y = Ax. Thus A is closed. From (2.10) and a standard result on Bochner integrals we get S(t)xn - xn = A / S(s)xn ds, Jo

t > 0.

Closedness of A implies JQS(s)xds £ domA and (2.8). In order to prove that dom A is dense we choose x £ X and set, for h > 0, 1 [h Xh = -r S(s)xds

£ domA.

For h I 0 we get Xh —> £• b) For /i > 0 we have i (S(fc) - J) /" S(i - s)f(s) ds = ^ f S(t + h- s)f{s) ds - - /

+

S(t + h-

s)f(s)ds

£s(t-8)(±(f(8+h)-f(s)))d8,

which for h | 0 implies (2.9). Here we have used that / ' is Bochner-integrable on any interval [0, T], T > t. Therefore Theorem A.5 for p = 1 and g = f implies l i m H 0 \f - fh\Ll(o,t;X) = 0. Observe that f'h(s) = h"1 j*+hf'(s) ds h-1(f(s + h)-f(s)),s£\0,t}. D For a linear operator A on X we define the powers An, n = 0 , 1 , . . . , recursively by A 0 = i" and domA™ = {x£ domA™"1 | An~1x £ domA}, Anx = A(An-1x),

xedomA™,

for n = 1,2,... . Concerning powers of infinitesimal generators we have the following results: 2.11. Proposition. Let A be the infinitesimal generator of the Co-semigroup S(-). Then the following is true: oo

a) ( | domA™ is dense in X. n=l

b) For all n = 1,2,... we have 5(t)domA™ c domA™, t > 0. Moreover, for all x £ domA™, the mapping t —»• S(t)x is in Cn(0,oo;X) D C(0, oo;domA") with (2.11)

T ^ * ) * = AnS(t)x

= S(t)Anx,

t > 0.

2.1. Examples and basic definitions

65

c) An is closed, n = 1,2,... . Proof, a) Let T> be the set of all scalar C°°-functions on [0, oo) with compact support in (0,oo). Then the integral f^°<j>(s)S(s)xds exists for all x G X and <j> G V. We set M = {f™(/){s)S{s)xds \ x G X, 0 G V}. For y = f^°(f)(s)S(s)xds G .M it is easy to see that /»oo

2/G domA

(j)'(s)S(s)xds. Jo Since $ G X>, we can continue by induction and get 2/GdomA"

and

Ay = -

Any = ( - 1 ) " /

and

^ ( n ) (s)5(s)a;ds,

ra=l,2,...,

which proves yVf c ( ~ d i domA™. Assume that M is not dense in X. Then, by the Hahn-Banach theorem, there exists a n i o G X \ M. and an XQ G X* such that (x 0 , XQ) = 1 and

(y, x%) = 0 for all y G .M.

Consequently we get for all 0 G V 0 = ([

4>{s)S(s)x,ds,x*0\= f <j)(s)(S{s)x,x*0)ds.

The function g(s) = (S(S)XQ, XQ) is continuous on [0, oo) with #(0) = 1. Therefore there exists an SQ > 0 with g(s) > 0 on [0,So]. We choose a non-negative 4>o G V with supp(/>o C (0, So). For 4>0 we have J^°(J)O(S)(S(S)XO,XQ) ds > 0, a contradiction. Thus statement a) is proved. b) Statement b) of the theorem is already established for n — 1 by Theorem 2.9. Assume that we have proved the result for k = 1 , . . . ,n and choose x G dom A™+1. Then we have Anx G domA, which by Theorem 2.9 implies jS{t)Anx

= AS{t)Anx

= S{t)An+1x,

t > 0.

n

In particular we have S(t)A x G domA, t > 0. Using x G domAn we get (dn/dtn)S(t)x = S(t)Anx = AnS(t)x G d o m A This shows S(t)x G dom A™+1, t > 0, and ^-—rS(t)x = ^-S(t)Anx dtn+L dt Observing AS{t)Anx = AAnS(t)x finished. c) We assume that A71-1 is closed Xj —> x and AnXj —> y as j —> j^~iS{t)xj

- An-xXj

= AS(t)Anx

= S(t)An+1x,

= An+1S(t)x

the proof for statement b) is

t > 0.

and choose a sequence (XJ) C domA™ with oo. Integrating (2.11) with x = Xj we obtain

= An-lS{t)Xj

- An-1xj

= f S(s)AnXjds,

t > 0.

66

Chapter 2. Linear

Semigroups

This implies An~1(S(t)xj — Xj) —> fQS(s)yds as j —• oo. Together with S(t)xj - Xj -+ 5(t)x - x and closedness of An~x we get 5(£)x — x G domA™ -1 and 5(t)i4 n _ 1 a: - An~1x = An-l(S{i)x - x) = JQtS{s)yds. This implies n 1 n 1 \imti.0(l/t)(S(t)A ~ x - A ~ x) = y, which proves An~1x G domA and AAn-lx = y. U

2.2.

Cauchy problems and mild solutions

In this section we discuss the following non-homogeneous linear Cauchy problem: (2.12)

| « ( t ) = A n ( t ) + /(*),

t>0,

w(0) = x0, where the operator A on X and the function / : [0, oo) -» X are given. 2.12. Definition. We assume that / G i, 1 oc ([0,oo);X). A continuous function u : [0, oo) —» X is called a strong solution of (2.12) if and only if (i) u(t) G domA a.e. on [0, oo). (ii)

Au(.)€Llc([0,oo);X).

(in) u(t) =x0+

Jo

Au(s) ds+

Jo

f(s) ds,

t > 0.

Instead of (ii) and (iii) we can equivalently require that (ii') «(•)€ < £ ( [ ( ) , oo); X), (iii') u(0) = x0 and -ru(t) = Au(t) + f(t) a.e. on t > 0. 2.13. Theorem. Let A be infinitesimal generator of a Co-semigroup S(-) and assume f G L^oc([0, oo);X). a) Let x0 G domA be given and assume that f G W££ ([0, oo); X). function u defined by (2.13)

u(t) = S(t)x0+

I S(t-s)f(s)ds, Jo

Then the

t>0,

is a strong solution of (2.12). Moreover, we have u G C(0,oo;domA). b) Assume that u is a strong solution of (2.12). Then u is given by (2.13).

2.2. Cauchy problems and mild solutions

67

Proof, a) Let u be defined by (2.13). From Theorems 2.9 and 2.10, b), we conclude that u € C(0, oo; dom A) and Au{t) + f{t) = Sit) iAx0 + /(0)) + / Sit - s)f'is) ds,

t > 0.

Jo

Integrating from 0 to t we get f

(AU(T)

+ fir)) dr=

Vo

f

S(T)(AC0

+ /(0)) dr + [ f

Jo

S{T

- s)/'(s) d s d r

v/o Jo

= I ^ (S(T)SO) dr + y 5(r)/(0) dr + /" 5(s) /

f'(T-s)dTda

= Sit)x0-xQ+

/S(r)/(0)dr Jo

+ [

Sis)ifit-s)-fiO))ds

Jo

= S(t)x0 -x0+

/ S{tJo

s)fis) ds = u(t) -x0,

t> 0,

i.e., u is a strong solution of (2.12). b) Let u be a strong solution of (2.12). For fixed t > 0 we set v(s) = 0 < s 0 we get i (v(s + h) - u(s)) = i (S(t - s - h)u(s + h)-

Sit—s)uis),

Sit - «)«(«))

= -i(5(i - (s + A))(u(s + ft) - «(«)) + j-S(t - a - h))[u(s) -

S(h)u(s)).

For h 4- 0 this implies - u ( i ) = S(* - s) (u'(s) - Au(s)) ds Analogously we get, for s e (0,t] and h £ (0,s),

a.e. on [0, i).

-—vis) = S(t - s) (u'(s) - Auis)) ds

a.e. on (0, t],

so that v'(s) = Sit - s) (u'{s) - Auis)) = Sit - s)fis) Integrating from 0 to t we get (2.13).

a.e. on (0, t). •

Chapter 2. Linear

68

Semigroups

Theorem 2.13, b), shows that strong solutions of (2.12) are unique, provided A is infinitesimal generator of a Co-semigroup and / € L,1oc([0, oo); X). Since J0S(t - s)f(s)ds is well-defined for / G .LI1oc([0,oo);.X"), the question arises about the connection between u given by (2.13) and the Cauchy problem (2.12). For xQ e X and / £ Ljoc([Q,oo);X) we can find sequences (x„)ne^ C dom.A and (/„)„ e N C W^([0,oo);X) with xn -> x0 and | / - / „ | z , i ( o , r ; x ) -> 0 for all T > 0 as TL —)• oo. The problems {djdt)un — Aun -f- fni un{0^ — xn have unique strong solutions (Theorem 2.13). This motivates the following definition. 2.14. Definition. Let A be a linear operator on X and / e £^([0,00);-X"). A continuous function u : [0,00) —¥ X is called a mi/d solution of (2.12) if and only if u(0) = £0 and there exist sequences ( / „ ) „ £ N C Lj oc ([0,oo);X), («„)„ e N C C(0, 00; X) such that un is a strong solution of (d/dt)un(t) = Aun(t) + fn(t) and lim | / - fn\mo,T;X) lim u«(£) = u(t)

= 0 for all T > 0, uniformly on [0, T\ for any T > 0.

u—•oo

The following theorem gives a characterization of mild solutions if the operator A is closed: 2.15. Theorem. Assume that A is a closed operator on X and that f £ Lloc([0, 00); X). Then a continuous function u : [0, 00) —• X is mild solution of (2.12) if and only if t

(2.14)

/ u(s) ds £ dom A, Jo

t > 0,

and (2.15)

u{t) = x0 + A / u(s) ds+ f(s) ds, Jo Jo

t > 0.

Proof, a) Assume first that u is a mild solution of (2.12) and choose sequences (/n)neN, (un)neN according to Definition 2.14. Since the un are strong solutions, we have JQun(s) ds € dom A, t > 0, and A / un(s)ds+ JO

/ fn(s)ds= io

/ (4u n (s) + /n(s))ds ./o = / u'n(s)ds = un(t)-un(0), Jo

t>0,

2.2. Cauchy problems and mild solutions

69

which implies lim A / un(s) ds = u(t) - xo - / f(s)ds,

™-+°°

Jo

t>0.

Jo

Obviously we have lim„_yoo J0un{s) ds = JQu(s)ds. Closedness of A implies J0u(s)ds G domA and AjQu(s)ds = u{t) - Xo — f0f(s)ds, t > 0, i.e., u satisfies (2.14) and (2.15). b) Assume that u satisfies (2.14) and (2.15). For ft > 0 we define uh(t) = -

u(s)ds

and

fh(t) - - /

f(s)ds,

t > 0.

The functions Uh are continuously differentiable on [0, oo) with (2.16)

^-uh{t) = ^(u(t + h)-u(t)), t>0. at n By standard results on Steklov means (see Theorem A.5 for p = 1) we have also, for any T > 0, ljm|/-/ft|ii(0,T;X) = 0 . From (2.14) we conclude that Uh G domA, t > 0. Using (2.15) and (2.16) we obtain rt+h

Auh{t) = — (A / = -(u(t

u(s) ds - A

u(s) ds)

+ h)-u(t))--J

= jtuh{t)

- fh(t),

f(s)ds t > 0,

i.e., Uh is a strong solution of (d/dt)u — Au + fh on [0, oo). The estimate \u(t) - Uh{t)\ < — /

)u(t) - u(s)\ ds <

max

\u(t) - u(s)\

proves limhio Uh(t) = u(t) uniformly on intervals [0,T], T > 0. Thus we have shown that u is a mild solution of (2.12). • 2.16. Theorem. Let A be the infinitesimal generator of a Co-semigroup S(-) and assume that f 6 L11oc([0, oo);X). Then u{t) = S(t)x0 +

S(tJo is the unique mild solution of (2.12).

s)f(s) ds,

t > 0,

70

Chapter 2. Linear

Semigroups

Proof. Let u be defined by the variation of constants formula. Then we have pt

rt

pt

I u(s)ds = / S(s)xods+ Jo Jo = [ S(s)x0ds+ Jo

i<s

/ / S{s Jo Jo f f Jo Jo

-T)f(T)d.Tds S(T)f(s)dsdr.

Using Theorem 2.10, a), we obtain J0u(s) ds £ dom A for all t £ [0, oo) and A f u{s) ds = S(t)x0 -x0+

[ (S(t - T)J(T)

Jo

= u{t)-x0-

- / ( r ) ) ds

Jo / Jo

f{r)dT,

which is (2.15). If «i, i«2 are two mild solutions of (2.12), then u = u\ — «2 is a mild solution of the homogeneous problem —u(t) = Au(t), t > 0, w(0) = 0. at According to the definition of mild solutions there exist a sequence (/„) C Lloc([0,oo);X) with l/nlL^o.T;*) -> 0 for all T > 0 and a sequence («n)neN of strong solutions of (d/dt)un = Aun + fn with lim„_>oo un(t) = u(t) uniformly on intervals [0,T], T > 0. Theorem 2.13, b), implies that un(t) = S(t)un(0) + J0S(t — s)fn(s)ds, t > 0, n = 1,2,... . Since we have u(0) = 0, we get lim„_>oo un(0) = 0, which implies u(t) = lim„_+0o un(t) = 0, t > 0. D Mild solutions can also be characterized as weak solutions. We recall that for a densely defined linear operator A on X and any element x* G X* there exists at most one element y* £ X* such that (Ax, x*) = (x, y*) for all x £ dom A. This justifies the following definition of the adjoint operator A* corresponding to a linear operator A with dom A = X: dom A* = [x* £ X* |there exists ay* G X* such that (Ax,x*) — (x,y*) for all x £ dom A], A*x*=y*,

x*edomA*,

where y* £ X* is the unique element with (Ax,x*) — (x,y*) for all x £ dom A. We shall need the following lemma: 2.17. Lemma. Let A be a closed operator on X with dom A = X and Xo,yo £ X such that (yo,x*) = (x0,A*x*)

for all x* £ dom A*.

Then we have xQ £ dom A and y 0 = Axo-

2.2. Cauchy problems and mild solutions

71

Proof. Let GA denote the graph of A, GA '•= {{x, Ax) e l x X j x G dom A}, and assume that (xo, yo) $. GA- Since GA is a closed subspace of X x X, the Hahn-Banach theorem assures the existence of two bounded linear functionals X-t , X X G X* such that (2.17)

{x, x\) + {Ax, x^) = 0 for all x G dom A,

and (2.18)

(x0,xl)

+

{y0,x*2}^0.

By definition of A* equation (2.17) implies x\ G dom A* and A*x% = —x\. Using this in equation (2.18) we get {XQ,A*XX) ^ (Uo,^), a contradiction. D 2.18. Theorem. Let A be a closed operator on X with dense domain and assume f G L11oc([0, oo);X). Then a continuous function u : [0, 00) —» X is a mild solution of (2.12) if and only if u is a weak solution of (2.12), i.e., u is continuous on [0,00) with the following properties: (i) w(0) = x0. (ii) For allx* G dom A* the function t —» (u(t),x*) is absolutely continuous on all intervals [0,T], T > 0, and (2.19)

j{u{t),x*)

= (u{t),A*x*) + {f{t),x*)

a.e. on [0,oo).

Proof, a) Let u be a mild solution of (2.12). For any x* G dom A* we get from (2.14) (A J*u(s) ds,x*) = (f*u(s)ds,A*x*), t > 0. Then (2.15) implies that, for all x* G dom A*, (u(t),x*) = {x0,x*) + (f

u{s)ds,A*x*\

+ (f

f(s)ds,x

= (x0, x*) + [ {(u(s), A*x*) + (f(s), x*)) ds, t > 0. Jo This implies that t —> (u(t),x*) is absolutely continuous on intervals [0,T], T > 0, and (2.19) holds. b) If u is a weak solution, then (2.19) implies u(t)-x0-

/ f(s)ds,x*\

= /

u(s)ds,A*x*

for all i > 0 and all a;* G dom A*. According to Lemma 2.17 we have J0 u(s) ds G dom A and A jQ u(s) ds = u(t) - x0 — J0 f(s) ds. • The following theorem shows that existence and uniqueness of mild solutions is also sufficient for a closed operator to be infinitesimal generator of a Co-semigroup.

72

Chapter 2. Linear

Semigroups

2.19. Theorem (Ball). Let A be a closed operator A on X and assume f G Lioc([0, oo);X). The operator A is the infinitesimal generator of a Co-semigroup S(-) if and only if the abstract Cauchy problem (2.12) has for any XQ G X a unique mild solution U(-\XQ). In this case u{t; x0) = S(t)x0 +

S(t- s)f(s) ds, t>0. Jo Proof. In view of Theorem 2.16 we have to prove that existence and uniqueness of mild solutions imply that A is infinitesimal generator. The assumptions imply that the homogeneous problem (d/dt)v(t) = Av(t),

(2.20) V ;

t > 0,

v(0) = x0

has a unique mild solution for any XQ G X, which we denote by v(t;xo). define the operators S(t), t > 0, by S(t)x0=v(t;x0),

We

t>0.

It is clear that 5(0) = / . In order to prove the semigroup property we choose xo € X and t > s > 0 and define ui(t) = v(t + s;xo) = S(t + s)xo and u-i(t) = v(t;v(s;xo)) = S(t)S(s)xo. Obviously, u
I Jo

v(T;xo)dr—

/ Jo

V(T;XQ)

dr G d o m A

Using (2.15) we get also /•t+s

Ui(t) = X0 + A

v(T;X0)dT Jo

— XQ + A / v(T;x0)dT Jo

+ A / v(T + s;x0)dT Jo

= v(s;xo) + A

ui(r)dT, t>0. Jo Theorem 2.15 implies that U\ is a mild solution of (2.20) with initial value v(s;xo)- By uniqueness of mild solutions we have U\ = U2, i.e., S(t + s) = S(t)S(s). Obviously, the function t —• S(t)x0 is continuous on [0, oo) for any x0 G X. In order to prove that the operators S(t) are bounded, we define for any T > 0 the mapping $ T : X ->• C(0, T; X) by ($Tx)(t) = v(t; x) = S{t)x,

t e [0,T],

xeX.

2.2. Cauchy problems and mild solutions

73

Let (xn)nen be a sequence in X with xn -> x and $T%n -> w G C(0, T; X) as n —>• oo. For any n there exists a unique mild solution v(-;xn) of (2.20) with initial value xn. By Theorem 2.15 we have f0($TXn)(s) ds G dom A and A I (Txn)(s)ds = Jo

($Txn)(t)

t>0,

n = 1,2,...,

which implies limn_>.oo Af0($TXn)(s) ds — w(t) — x, 0 T%- Therefore $ ^ is a closed operator defined on all of X. By the closed graph theorem §T is bounded and consequently ||5(-)|| is bounded on any interval [0,T]. Thus £(•) is a Cosemigroup on X. Let B denote the infinitesimal generator of S(-). We have to prove that A = B. We first choose x £ dom B. Then we have (2.21)

lim ~ (S(h)x - x) = Bx.

On the other hand v(-) = S(-)x is the unique mild solution of (2.20) with i/(0) = x. By Theorem 2.15 this implies j - (S(h)x -x)=Aj-

f S(s)x ds,

h>0.

This together with (2.21) implies limhi0 A^h-1 JQ S(s)xds) = Bx. Furthermore, we have lim/40 h~1 JQ S(s)xds = x. Thus closedness of A implies x G dom A and Ax = Bx, i.e., A is an extension of B. Let now x G dom A. Then S(t)x and S(t)Ax are mild solutions of (2.20) and consequently, by Theorem 2.15, (2.22)

A

S(s)xds

= S(t)x-x

and

A / S(s)Axds

Jo

for all t > 0. Since s —> JQS S{r)Axdr A is closed, we have

n

is a continuous mapping into dom A and pz

s

(2.23)

= S(t)Ax - Ax

Jo

S(T)Axdrds=

ps

A S{r)AxdTds Jo Jo = / (S(s)Ax - Ax) ds, Jo

t > 0.

74

Chapter 2. Linear

Semigroups

We define pt

ft

nt

z{t):=

/ S(s)Axds-A / S(s)xds = / S(s)Axds - S(t)x + x, t > 0. 7o Jo Jo Obviously, z{t) is continuous on t > 0 with z(0) = 0. Using (2.22) and (2.23) we get ft

rt

pS

rt

A I z(s)ds = A / / S{r)AxdTds-A Jo Jo Jo

Jo

= f (S(s)Ax -Ax)ds-A Jo

ps

A

Jo

S(r)xdTds

f (S(s)x - x) ds Jo

= / S{s)AxdsA I S{s)xds = z(t), t>0. Jo Jo According to Theorem 2.15 this proves that z(t) is a generalized solution of (2.20) with initial value z(0) = 0. By uniqueness of generalized solutions we have z(t) = 0 on t > 0 and consequently -(S(t)x-x)=t For 11 0 we get x € dom B.

2.3.

f S(s)Axds, t Jo

t>0. D

The Hille-Yosida theorem

In view of the results of the previous section it is important to give a characterization of infinitesimal generators of Co-semigroups. We first consider necessary conditions. 2.20. Proposition. Assume that A is the infinitesimal generator of a Cosemigroup S(-) of class G(M,UJ). Then Re A > u implies A € p(A) and /»oo

( A 7 - ^ l ) - 1 a ; = / e~xtS{t)xdt, x G X. Jo Proof. Let Re A > w and define the bounded linear operator R on X by (2.24)

/»oo

e~xtS{t)xdt,

Rx=

i £ l

Jo Then, for any x G dom A, rOO

t>00

xt

R(\I-A)x=

e- S{t)(\I-A)xdt = x-

lim e~xtS(t)x t—VOO

= - I = x.

1

—(e-XtS{t)x\dt

2.3. The Hille-Yosida theorem

75

On the other hand, for any t > 0, we define Rtx = I e~XrS(T)xdT, xeX. Jo It is easy to see that e~x'S(-) is also a Co-semigroup and its infinitesimal generator is A — XI. Therefore we have (using Theorem 2.10, a)) (XI - A)Rtx = (XI -A)f Jo = x- e~xtS(t)x,

e'XTS(r)xdT xeX.

This implies \imt-i.oc(XI—A)Rtx = x. On the other hand we have lim^oo Rt% = Rx, so that by closedness oi XI - A we obtain Rx e dom(A7 - A) = domA and (XI - A)Rx = x, x e X. This finishes the proof of R = (XI - A)~l. D 2.21. Corollary. Let S(-) be a Co-semigroup of class G(M,w) with infinitesimal generator A. Then {X 6 C | Re A > u>} C p(A) and

(2.25)

|| ( A / _ J r l <_JL_

for Re A > w and n = 1,2,... . Proof. From (2.24) we get, for any x £ X and any A with Re A > w, /»oo

(XI-A)-2x=

/»oo

e~XtS(t) Jo

e~XTS(T)xdTdt Jo

x( t+T

= / / e- - ^S(t Jo Jo

+ T)xdTdt=

/ / Jo Jt

e-XsS(s)xdsdt

xt

te~ S(t)xdt. I By induction we arrive at 10

Jo

(XI - A)~nx =

1

f°°

/ tn-le~xtS(t)x dt (n-l)l J0 for Re A > u>, n = 1, 2 , . . . . Repeated integration by parts gives \(XI-A)-nx\

Irl

f°°

< . ' ' ., / (n-iy. Jo M\x\ (ReA-w)n

tn-1e-tReXMeutdt for Re A > w.

The main result of this section is the following theorem: 2.22. Theorem (Hille—Yosida). Let A be a linear operator on X and M > 1, u> £ R be constants. Then the following two statements are equivalent:

76

Chapter 2. Linear Semigroups

(i) A is the infinitesimal generator of a Co-semigroup of class

G(M,u).

(ii) A is a closed, densely defined operator with (w, oo) C p(A) and (2.26)

\\{\I-AYn\\<

^

^

forX>w,n

= l,2,...

.

Proof. That (i) implies (ii) is clear from Theorem 2.10, a), and Corollary 2.21. The proof that (ii) implies (i) will be carried out in several steps. Step 1 (Resolvent): As in Chapter 1 we define the resolvent Jh of A by (compare (1.13)) Jh = {I-hA)-1,

Q
For x S dom A we have Jhx - x = (I - hA)~x(x -x + hAx) = (—1 ~ A)

Ax.

This together with (2.26) for n = 1 implies, for 0 < h < l/\u\, \JhX-x\<—

\Ax\ = \Ax\, xedomA, l/n — u> 1 — nu> and consequently lim/40 JhX = x for all x € dom A. Using (2.26) again we see that, for 0 < h < l/(2|u>|),

Thus we have shown (using dom A = X) (2.27)

lim Jhx = x

for all x e X.

hlO

Step 2 (Yosida approximation): The Yosida approximation A^ of A is defined by (compare (1.13)) Ah=AJh

= -(Jh-l),

Q

For x £ dom A we have AJ^x = JhAx. This and (2.27) imply (2.28)

lilim AhX = lim JhAx = Ax, h

HO

x e dom A.

HO

From (2.26) we get M

^(1

+

T^)'

4 f c ll
o
2.3. The Hille-Yosida

theorem

77

Using the last inequality in eAht = e~%lh exp f | Jh j , t > 0, we have the estimate

(*»)

|| e -|| S Me-./«f;I(i)*( T ^-)'.Me,p( r ^- t )

for 0 < h<

1/\LJ\, t > 0.

Step 3 {Convergence of exp(Aht)): From the representation eAnt

_ eAht

fd^s^t-s^

=

ds

Jo ds

= f e A f c 3 e y l ^ t ->(A h - Ah) ds,

t > 0, 0 < h, h< l/|w|,

and (2.29) we get the estimate \eAhtx - e^'a;| < MHe^^AhX

- A^x]

for t > 0, 0 < h, h < l/(2|w|). This estimate together with (2.28) shows that, for any T > 0, there exists a constant CT such that, for all x e X, \eAhtx - eA^x\

< cT\Ahx - A~hx\,

0
0
l/(2|w|).

For arbitrary i £ l w e choose (xn) C domA with \x — xn\ —>• 0 as n —• oo. Then we get using also (2.29) l e ^ ' z - e^'a:| < (||e Afct || + ||e^<||)|z - xn\ + \eA^xn < 2 M e 2 M 7 > - xn\ + cT\Ahxn for all t£[0,T\,h,h£ for all x £ X

-

-

eA^xn\

Ahxn\

(0, l/(2|w|)) and n = 1,2... . This estimate shows that

(2.30)

limeAhtx

S(t)x =

exists uniformly on [0,T] for any T > 0. From this it is also evident that S(-) is a Co-semigroup. The estimate (2.29) shows that S(-) is of class G(M,w). Step 4 (A is infinitesimal generator): Let B denote the infinitesimal generator of S(-). For any A > w, T > 0 and x e X we obtain, using also the estimates (2.29) and ||5(t)|| < Me"', t > 0,

\{\I-Ah)-lx-{\I-B)-1x\

= /

e~xt[eAht -

S(t))xdt

Jo

fT < / Jo

e-xt\eAhtx-S{t)x\dt

+ M W /T°°(exP(-(A - ^

t

)

+ c-^-)* ]A

78

Chapter 2. Linear

Semigroups

for h sufficiently small (i.e., h € (0, l/|u;|) such that A > w(l — hw)~l). and (2.30) immediately imply lim(A7 - Ah)~lx

(2.31)

This

B)~lx

= (XI -

for all x G X and A > w. On the other hand we have (2.32)

(XI - Ah)-lx

- (XI - A)'lx

= (XI - Ah)-l(Ah

- A)(XI -

A)^x

for all x € X, A > u and h sufficiently small. The representation (XI — Ah)-1 — f0°°e~xteAhtdt together with the estimate (2.29) implies ||(AJ - Ah)-l\\

w < M(X ) \ 1 — flu )

for h sufficiently small.

Therefore we get from (2.32) and (2.28) lim(AJ - Ah)-lx

= (XI - A)~xx,

This and (2.31) prove (XI - A)'1 A = B.

X > u, x e X.

= (XI - B)"1 for all A > w which shows D

2.23. Theorem (Exponential Formula). Let A be the infinitesimal generator of the Co-semigroup S(-) on X. Then we have for all x € X S(t)x = lim (I - -A) n

H

x,

t > 0,

the limit being uniform with respect to t in bounded intervals. Proof. For some constants M > 1 and w £ M we have S(-) € G(M,u>). Without restriction we can assume u > 0. For t > 0 and x e dom A we have S(t)x - { i - -A) ~nx = S(t/n)nx

(2.33)

J2 S(jt/n)

( / - -Aj

U J

- J2 S(jt/n)(l--A)~n

- ( i - -A) (s(t/n)x

~nx - ( / - -AJ

' ( 2 ( S ( t / n ) x - x) -

a)

S(t/n)Ax),

3=0

where we have also used that S(t) and (XI —A) From (2.26) we get /

7

t

A

\~n

x

commute for t > 0, A G p(A).

M

l( -n ) hjl^Jnr^

-1,2,....

2.4. Lumer-Phillips

theorem

79

Using this in (2.33) we have -n+j

S(jt/n)(l-±A)

<M2ejtbJ/n{l-ojt/n)-n+j

(2.34)

< M ^ ( l - ^ ) " "

forO
Theorem 2.9 implies n -(S{t/n)x-x)-S(t/n)Ax t n rt/n < -

/

t Jo < 2 max

n /"*/" = - / S(T)AndT t Jo

\S{T)AX

- Ax\ dr + \S{t/n)Ax

\S(T)AX-AX\,

X

-

S(t/n)Ax

Ax\

G d o m A 0 < t < T.

0
i

-

-

From this and (2.33), (2.34) we conclude S(t)x-

(^--^)

l 1 "a < 2M Te"

(1 - ojT/n)-n

max

|5(r)Ac - Ax\

0
for ar G dom^l and 0 < t < T. This proves that, for x G dom.4, we have

(

lim I

t n

\ ~n A) x = S(t)3 J

uniformly on [0,T]. Since ||(7 - {t/n)A)-n\\ < M{1 - LoT/n)-n < Me2"T t G [0, T] and n > 2wT, a standard density argument finishes the proof.

2.4.

for •

The Lumer-Phillips theorem

We recall the definition of a dissipative operator on X given for the general case in Definition 1.5. A linear operator A on X is dissipative if and only if for all x G &omA there exists an x* G Fx such that Re(Ar,a;*) < 0 or, equivalently (see Corollary 1.7), if and only if \(I — XA)x\ > \x\ for all A > 0 and x G domA, which in turn is equivalent to \(XI — A)x\ > \\x\ for all A > 0 and x G d o m A If X is a Hilbert space with inner product (•,•), then A dissipative means Re( Ax, x) < 0 for all x G dom A Let u g l b e given. Then it is easy to see that A — wl is dissipative if and only if for any x G dom A there exists an x* G Fx such that Re(Ax,x*) < u\x\2 which is equivalent to |(A7 - A)x\ > (A - u)\x\ for all A > u> and x G d o m A In a Hilbert space this is equivalent to Re(Ar, x) < w\x\2. It is also easy to see that m-dissipativity of A — ail is equivalent to m-dissipativity of A

80

Chapter 2. Linear Semigroups

2.24. Example. Consider the linear operator A defined by dom A = {0 G X |0 is absolutely continuous on (0,1) (2.35)

with 0(0) = 0 and 0' G X} A = -(f>,

0 £ d o m A,

where X = L p (0,1), 1 < p < oo, or X = C(0,1). Case i : X = L p (0,1), 1 < p < oo. Given 0 G X it is easy to see that * = |0|£p P |0| p ~ 2 0 G L 9 (0,1), 1/p + 1/g = 1, and (0,<£*) = | 0 | | P , i.e., 0* = F 0 . Using (d/dx)|0| p = p | 0 | p " 2 0 0 ' for 0 G dom A we get (A/>,0*> = - i - - J = a / (d/dx)|0(x)| p dx = - I _ L j | ^ ( i ) | P < 0. Hence A is dissipative on X. Case 2: X = C o (0,1) = {0 G C(0,1) | 0(0) = 0}. For <j> € X suppose that \\c = |0(zo)| for some xo G (0,1). Using 0'(xo) = 0 we get, for all A > 0, |A0 - A<j)\c > |A0(xo) + 0'(x o )| = A|0(x o )| = A|0| c . If |0|c = 10(1)1, then we have 0(1)0'(1) > 0 and consequently |A0 - A4>\c > |A0(1) + 0'(1)| > A|0(1)| = A|0| c ,

A > 0.

Finally, for | 0 | c = |0(O)| we have 0 = 0. Thus also in this case A is dissipative onX. Case 3: X = L 1 (0,1). For 0 G X we set 0* = |0| L i sign0 G L°°(0,1). It is easy to see that 0* G F<j>. For e > 0 we define ^

signr 1 / I r/e

if \T\ > e, -fi 1 ^ it \T\ < e.

For 0 G dom A we get / V,(<MzM(z) <*x = / JO

Ve(r) dr = # £ (0(1)) > 0,

J(t>(0) '0(0)

where r|-e/2

* « M = < i , T\,22 2e

if|r|>e, if |r| < e.

2.4. Lumer-Phillips

theorem

81

By Lebesgue's dominated convergence theorem we have / t/je((j)(x))(t)'(x)dx -» / sigii(f>(x)(f)'(x)dx Jo Jo

as e 4. 0,

so that

( A ^ * ) = H^|L1limtfe(
Therefore yl is also dissipative on X in this case. The equation (XI - A)(f> = v), ip e X, <j> e domA, A > 0, is equivalent to 0' = -A + ,

0(0) = 0,

resp. to (2.36)

-F

{x)= I e-x{x-T)tp(T)dT,

0 < x < 1.

Jo by (2.36) is in dom A tor X = Lp(0,1), 1 < p < 00, It is easy to see that <> / given and also for X = C'o(0,1). Thus A is m-dissipative.

2.25. Theorem (Lumer-Phillips). Let A be an operator on X and w € R. TTien i/ie following statements are equivalent: (i) yl JS i/ie infinitesimal generator of a Co-semigroup of class G(1,OJ) on X. (ii) A is densely defined and A — ul is m-dissipative. If X is reflexive, then (ii) can be replaced by (ii') A — UJI is m-dissipative. If (i) or equivalently (ii) is satisfied, then A — wI is strictly dissipative, i.e., we have Re(Ax,x*) < w\x\2 for all x e dom A and all x* G Fx. Proof. Without restriction we can assume that w = 0 (otherwise we replace A by A — ul). Note that A generates a semigroup of class G(l, u>) if and only if A — u)I generates a semigroup of class G{\, 0). a) Assume that (ii) is true. Dissipativity of A implies (2.37)

\{XI-A)x\

> X\x\,

xedomA,A>0.

For A > 0 we have range(A7 — A) = X. By the Open Mapping Theorem we conclude that (XI - A)'1 e C(X), A > 0. Then (2.37) implies

IKAJ-ii)-1!!^, A > O . The operator A is closed, because XI — A = ((XI — A ) - 1 ) - 1 is closed. By the Hille-Yosida theorem we see that A is infinitesimal generator of a Co-semigroup 5(-) of class G( 1,0).

82

Chapter 2. Linear

Semigroups

b) Conversely let A be the infinitesimal generator of a Co-semigroup of contractions. Then dom A = X by Theorem 2.22. For x* G Fx we have \{S{t)x,x*)

< \S(t)x\ \x*\ < \x\ \x*\ = \x\2,

t > 0.

This implies R e / i ( S ( t ) x - x), x*\ = -(Re(S{t)x,x*) - \x\2) < 0,

t > 0,

and by taking t 4- 0 Re(Ax, a;*) < 0 for all x G dom A, x* G Fa;. This proves that A is strictly dissipative. c) Assume that X is reflexive and A is m-dissipative. Let xJ5 G X* be such that (a;,a;J) = 0 for all x G dom A For the Yosida approximations Ah we get (2.38)

{Ahx,Xo) = - ( ( 7 - f t A ) _ 1 x - x , x 5 ) = 0,

x G dom A, ft > 0.

From Theorem 1.10, (hi), we conclude that (2.39)

lim(J - hA)~lx

= x,

|_Ahx| < \Ax\,

x G dom A, ft > 0.

x e dom A,

and

Since X is reflexive, it is weakly sequentially compact (see for instance [Yo, p. 126]), i.e., for any x G dom A there exist a sequence (ftn)neN with hn I 0 and a z € X such that (2.40)

w- lim Ahnx = w- lim A(I - hnA)~lx

= z.

As under part b) of this proof we see that A is closed. By Lemma 2.26 from below the operator A is also weakly closed (i.e., A is closed in Xw x Xw). Consequently, from (2.39) and (2.40) we conclude z — Ax, i.e., w-linin^oo Ahnx = Ax which together with equations (2.38) and (2.40) implies (AX,XQ)

— lim (Ah„x,XQ) — 0. n—>oo

By choice of XQ this implies (x — Ax,

ZQ)

= 0 for all x G dom A

or, equivalently, (y, XQ) = 0 for all y G range(7 - A) = X. This proves xg = 0, i.e., dom A = X. • 2.26. Lemma. Let A be a closed linear operator on A. Then A is also weakly closed.

2.5. A second order equation

83

Proof. The graph GA of A is a linear subspace of X x X. Then the result follows from a standard result on convex sets in Banach spaces (more generally in locally convex linear topological spaces; see for instance [La, p. 243]). • 2.27. Theorem. Let A be a closed densely defined operator on X. If A and A* are dissipative, then A is m-dissipative and thus the infinitesimal generator of a Co-semigroup of contractions. Proof. Let y G range(7 — A) be given. Then there exists a sequence (xn)nepi c dom A such that yn := xn - Axn ->• y as n -» oo. By dissipativity of A we get \x„\ < \yn\, n = 1,2,..., and \Xn

Xm\

S

\\Xn

XJJI)

J\\Xn

XTnJ\

— 12/TT.

2/m|)

771, 71 =

1, 2, . . .

.

Hence (xn)n^n is a Cauchy sequence in X. We set x = limn-^oo xn. From xn —> x and yn -» y we infer that also lim„_>.00 Axn — x — y. Since A is closed, we see that x G dom A and Ax = x — y, i.e., y G range(7 — A). Thus range(7 — A) is closed. Assume that range(7 — A) C X. Then there exists an x* G X*, x* ^ 0, such that ((/ - A)x, x*) = 0

for all x e dom A.

By definition of the adjoint operator this implies x* e domvl* and (I—A)*x* = x* - A*x* = 0. Dissipativity of A* implies \x*\ < \x* -A*x*\

=0.

This contradicts x* ^ 0 and proves range(7 — A) = X, i.e., A is m-dissipative. The result follows now from Theorem 2.25. •

2.5.

A second order equation

Let V C> H Q V* be a Gelfand triple as explained in Section 3.1. We shall apply the Lumer-Phillips theorem to the following second order equation in variational form: (2.41)

p(,«'(*)) +a(,u(t)) = (4>,f(t)) = (&/(*))

H

a.e. on t > 0 for all 4> G V. Here / is a function [0, oo) ->• H and p : H x H —^ C, /i,CT: y x y —>• C are sesquilinear forms2, where p and a are hermitean. 2

We follow here the convention that sesquilinear forms are linear in the first argument and conjugate linear in the second argument.

84

Chapter 2. Linear

Semigroups

Moreover, we assume that, for non-negative constants Mi, M2, M3 and mi, 1713, we have \P(,II>)\<MI\\H\II>\H, (2

(

0e#>

P(^^)>"»I|0IH.

'

'

,*p£H,

Reti(4>,4>)>0, £V, \,il>)\<M3\\v\rl>\v, , , ,, a ( 0 , < £ ) > m 3 | < ^ , 4>GV.

(2.42c) V

W ^ ,

From (2.42a) and (2.42c) it follows that (2.43)

({h,A),{h,ip2))x

:=i,2) +

PWi,ih)

for ((j>i, ijji) G V x H, i = 1,2, defines an inner product on X = V x if such that •I try

the associated norm \(, <j>)+p(tp, ip)) , ((/), ip) G X, is equivalent to the standard norm on X (i.e., to |(<£,VOI = (l^lv + M J J ) )• The given sesquilinear forms p, /J, resp. a define bounded linear operators Mo : H —> H, Do : V ->• V* resp. A0 : V ->• F* by (compare Theorem 3.2) {4>, MoV) = {4>, M o V k = />(& V0> (2.44)

4>,^eH,

<& Anfr) = i*{4>, */>), , 1> e V,

The operators Mo and A0 are invertible by (2.42a) and (2.42c). For a function u : [0, oo) -> V with «'(i) G V and «"(<) € if a.e. on i > 0 equation (2.41) is equivalent to (2.45)

M0u"(t) + D0u'(t) + A0u(t) = /(*)

a.e. on i > 0.

This equation implies also Aou(t) + Dou'(t) G if a.e. on t > 0. We define the operator A on X by d o m A = {(,ip) GX I V G V and AO^ + AJV' G i f } , (2-46)

,

=

f

0

\-M^Ao We set a;(i) = (u(t), u'(t))J, (2.47)

Hi?

-MoXD0,

i G [0, oo). Then equation (2.45) is equivalent to

—x{t) = Ax(t) + f(t)

where / > ) - (0, M 0 _ 1 /(*)) T , t > 0.

a.e. on t > 0,

2.5. A second order equation

85

2.28. Lemma. Let the operator Abe given by (2.46). Then A is m-dissipative on X (equipped with the inner product (2.43)). Proof. We choose x = (, VO G d o m A Using (2.44) we get (a:, Ax)x = a{4>, V>) + p(V>, -M o _ 1 (i4o0 + A)V0) =

- (, A)V>> = -2ilm(tP,A04,)

- (V, A)V>>,

which in view of (2.42b) implies Re(x,Ax)x

= -Re(ip,D0ilj}

— -Re/i(V>,V0 < 0,

i.e., A is dissipative. In order to prove that range(A7 — A) — X for some A > 0 we consider the equation (2.48)

(A/-A)(W) = (s,/i)6l,

(^,V)GdomA

This equation is equivalent to Ac/> — tp — g and Xtp + MQ1(AQ4> + DQIJJ) = h resp. to ( A }

\{X, M0tf>) + (X, A0 + D0i>) = (X, M0h)

for all X € V.

The second equation can be written as

Ap(x. V>) + ^(x, 4>) + M(X, VO =

P(X> &),

x e V.

Using the first equation of (2.49) we get (2.50)

MxA)=p(x,h)+\p{x,9)+v(x,9),

X^V,

where the sesquilinear form v> : V x V —• C is defined by v\ (x, <£) = A2p(x, ) + AM(X, 0) +
X,£ V.

The sesquilinear form v\ is bounded and satisfies the estimate K ( X , X ) | > mslxlv " A ' M i l x ^ - AM 2 | X |^ > ™M2v ~ {\2c2M1 + \M2)\x\2v,

X&V,

where c is the embedding constant for the embedding V C> H. This shows that, for some Ao > 0, we have

K(x,x)l> 2TO3lxlv, x e v ,

O
According to the Lax-Milgram theorem (see Theorem 3.2) there exists a bijective bounded operator B : V —• V* with bounded inverse such that v\(x, 4>) = (X,B4>), xA^V- This and (2.50) imply (j> = B-1(Moh

+ \M0g + D0g) € V,

0 < A < A0.

86

Chapter 2. Linear

Semigroups

Since g and

+ DQIJ) G H. Thus (0,-0) G dom A and range(A7 - A) = X forAe(0,A 0 ]. D Observing that X as a Hilbert space is reflexive we get from the LumerPhillips theorem: 2.29. Theorem. The operator A defined by (2.46) is the infinitesimal generator of a contraction semigroup S(-) on X. If x(0) G domA and / G ^ ( [ O . o o ) ; ^ ) , i.e., / G W^([0, o o ) ; * ) , then x(t) = S(t)x(0) + J0S(t — T)f(r)dT, t > 0, is the unique strong solution of equation (2.47). For x{t) = (u(t),v(t))T equation (2.47) is equivalent to u'(t) = v(t) and v'(t) = - M 0 - 1 ( ^ 0 M ( * ) + Dov{t)) + M 0 _ 1 / W , t > 0. By Theorem 2.13, a), we have also x G C(0, oojdomA), i.e., x G C(0, oo;X) and Ac G C(0, oo; X). Therefore we have u G C^O, oo; V) n C 2 (0, oo; # ) and M0u"{t) + D0u'(t) + A0u(t) = f(t)

a.e. on * > 0.

This is equivalent to p((j), u"{t)) + n{4>, u'(t)) + a(<)>,u{t)) = {, f(t))

a.e. on t > 0

for all 4> G V. If o;(0) G V x H and / G £^.([0, oo); # ) , then x{t) = S{t)x{0) + J*S(t T ) / ( r ) dr, t > 0, is the unique mild solution of (2.47). From Theorem 2.15 we get, for x{t) = (u(t),v(t))T and t > 0, u{t) = u(0) + / (2.51)

V(T)

dr,

t t t v(t) = v{0) - MQ1 (A0 f U{T) dr + DO f V(T) d r ) + f M^f{r)

dr.

The first equation implies u G C(0, oo; V)nC 1 (0, oo;H). Using A0 JQU(T) dr = J0Aou(r)dT and J0v(T)dr = u(t) — u(0) we obtain from the second equation of (2.51) (X,M0u'(t))

= (x,Mov(0))

- (X, f

A0u(r)dT)

Jo

-(X,Do(u(t)-u(0)))

+ (x, [

f(r)dr)

Jo

= {x, M0v(0)) - f (x, A0U(T)) Jo

dr - (X, D0u(t))

+ (X,D0u(0))+ f (x,f(T))dr Jo

2.5. A second order equation

87

a.e. on [0, oo) for all x £ V. This equation is equivalent to p(X, A*)) + M(X, u{t)) + / a(X,

«(T))

dr

Jo

= P(X, «'(0)) + M(X, «(0)) + / (x, /(r)> dr Jo a.e. on [0, oo) for all x € V. Differentiating we get - (p( X , «'(*)) + M(X, «(*))) + *(x, «(<)) = (X, /(*)> a.e. on [0, oo) for all x £ ^ , which is a weak form of (2.41).

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CHAPTER 3

Analytic Semigroups In this chapter we present the basic facts on analytic semigroups resp. on linear Co-semigroups on Eg which can be extended analytically to a cone in C containing RQ~. In particular we prove that an operator is the infinitesimal generator of an analytic semigroup if and only if it is sectorial (Section 3.2). In Section 3.1 we describe the construction of sectorial operators based on the concepts of Gelfand triples and F-coercive sesquilinear forms together with an application of the Lax-Milgram theorem. Our presentation follows [Pal] resp. [Ta].

3.1.

Dissipative operators and sesquilinear forms

In this section we present a construction for a special class of m-dissipative linear operators on Hilbert spaces, the so-called sectorial operators. These operators play an important role in the context of parabolic partial differential equations. For a complex Banach or Hilbert space Y we denote by Y* the antidual space of all bounded antilinear functional (called also conjugate linear junctionals) on Y1. If Y is a complex Hilbert space with inner product (•, -)y then, according to Riesz' theorem, we can identify Y* with Y by associating with x € Y the antilinear functional y —¥ (x,y}y, y sY, which gives a linear isometry Y —»• Y*. If we associate with x £ Y the linear functional y —> (y,x)y, y G Y, then we get an antilinear isometry between Y and the dual space of all bounded linear functionals on Y. We shall use also that a complex Hilbert space Y is antireflexive, i.e., for any x** € Y** there exists a unique x € Y such that x**(y*) — y*(x) for all y* € Y*. The mapping defined by x** —>• x is a linear isometry Y** -¥ Y. Let if be a complex Hilbert space and V C H be a complex antireflexive Banach space. 1 f G Y" satisfies | / ( y ) | < const. |j/|y for all y G Y and f{ay1+f3y2) for all a, /? €C and 3/1, y2 G Y.

89

=

af(y1)+Pf(y2)

Chapter 3. Analytic

90

Semigroups

3.1. Lemma. If the embedding V C» H is dense and continuous, then this is also true for the embedding H Q V*. Proof. Let M 0 be the embedding constant for V Q H, i.e., \V\H < MQ\V\V, v € V. Then for any h G H we have \{h,v)H\

< \h\H\v\H < M0\h\H\v\v,

v€V,

i.e., \h\v < Mo\h\n- Thus the embedding H Q V* is continuous and Mo can also be chosen as embedding constant for the embedding H Q V*. Assume that H is not dense in V*. By the Hahn-Banach theorem there exists a « £ V** = V, v ^ 0, with v(h) = {h,v)H = 0 for all he H. This implies v = 0, a contradiction. O We call the triple (V, H, V*) a Gelfand triple with pivoting space H. In a Gelfand triple the anti-duality pairing x*(v), v € V, x* G V*, is the continuous extension of (-,-)// restricted to i? x V. In particular, for any x* G V* there exists a sequence (hn) C if with x*(v)=

lim (hn,v)H,

veV,

the limit being uniform for w in bounded subsets of V. We already assumed V C H. By the identification H — H* the elements of H represent bounded antilinear functionals on H (to h G H there corresponds the functional w —> (h,w)). In view of the continuous embedding V Q H the restriction of such a functional to V is a bounded antilinear functional on V. In this sense we consider H as a subspace of V*. Therefore the embeddings V Q H, V Q V* resp. H Q V* are given by the restriction of the identity mapping to the corresponding subspace. If the Hilbert space H is real, then we have the same construction as above. We only have to delete the prefix 'anti' everywhere. A mapping b : HxH —^ C is called a sesquilinear form on H if y -> b(x, y), y G H, is antilinear for all fixed x G H and x —> b(x,y), x G H, is linear for all fixed y G H. A sesquilinear form &(•, •) is called bounded (or continuous) if there exists a constant 7 > 0 such that (3.1)

\b(x, y)\ < i\x\H\y\H,

x,y G H.

The norm of &(•, •) is defined as 111.11

\\b\\ =

SU

Kxiv) P

I ,

1 ,1

•

The following theorem is of central importance when solving Dirichlet problems for elliptic equations.

3.1. Dissipative operators and sesquilinear forms

91

3.2. Theorem (Lax-Milgram). Let b : H x H -» C be a sesquilinear form satisfying (3.1) and (3.2)

\b(x,x)\>6\x\2H,

x£H,6>0.

Then there exists a unique linear bijective operator B : H —• H* which is continuous in both directions such that b(x, y) = {Bx){y),

x,y G H.

l

Moreover, we have ||B|| < 7 and \\B~ \\ < 1/8. Proof. For fixed x G H the mapping y —> b(x,y) is antilinear and bounded (by (3.1)). Therefore there exists a unique w 6 H* such that b(x,y) = w(y), y £ H. We define Bx — w. It is easy to see that B is linear. From (3.1) we obtain \{Bx){y)\ = \b(x,y)\ < -y\x\H\y\H,

x,y G H,

which implies \BX\H* < 7|a;|if, i.e., B is continuous with ||B|| < 7. Assume that Bx = 0. Then (3.2) implies 0 = |(5a;)(a;)| = \b{x,x)\ > 8\x\2H, i.e., x = 0. Thus B is injective. The estimate 6\x\*H < \b{x,x)\ = \(Bx)(x)\ < \x\H\Bx\H.,

X G H,

1

implies <5|a;|^ < \BX\H*, X G H, or equivalently \B~ W\H < (1/S)\W\H*, W G range B, i.e., B^1 is continuous with ||JB_1]| < 1/8. It remains to prove surjectivity of B. Continuity of B~l and B implies that range B is closed. Suppose that range B ^ H*. By the Hahn-Banach theorem there exists a yQ G H, yQ ^ 0, such that (Bx)(y0) = 0 for all x G H. This implies for x — yo 0 = \(By0)(yo)\ =

\b(y0,yo)\>S\y0\2H

and consequently yo = 0. This contradiction proves range B = H*.

•

For the rest of this section we denote the norm and inner product on H by || • || and (-, •), respectively, and assume that V C H C V* is a Gelfand triple. 3.3. Definition. A sesquilinear form a : V x V —> C is called V-coercive if there exist constants 7,8 > 0 and K £ R such that (3.3)

|o(a;,|/)|<7|a;|v|»|v,

x,y€V,

and (3.4)

Rea(x,x)>6\x\lr-K\\X\\2,

X G V.

The inequality (3.4) is usually called Gdrding's inequality. If (3.3) and (3.4) hold with K = 0, then a(-, •) is also-called V-elliptic.

92

Chapter 3. Analytic

Semigroups

3.4. Theorem. Let a(-, •) be a V-coercive sesquilinear form. Then the following is true: a) There exists a unique linear operator A on V* with dom^4 = V such that, for any w E V, (3.5)

a(v,w) = -(Av)(w),

w£V.

Moreover, \\A\\c(v,v) < 1> where 7 is the constant in (3.3), and A — KI is a bijective mapping V - J F * with \\(A — Kl)~1\\crv,v) < V^> where 5,K are the constants in (3.4). Moreover, \\(A—KI)~1\\C(V,V-) ^ MQ/5, i.e., K G p{A), where M0 is the embedding constant for V c> H. b) Let A be the part of A in H, i.e., domyl = {x £ V \ Ax G H} and Ax = Ax for x G dom A Then the following is true: (i) a(v,w) = —(Av,w), v G dom A, w EV and dom A = {v G V I there exists an a > 0 such that \a(v,w)\ < a\\w\\ for allw G V}. (ii) dom A is dense in H. (iii) K G p{A). (iv) A — KI is m-dissipative. Proof, a) For fixed v G V the mapping w —» a(v,w), w G V, is a bounded antilinear functional / G V*. We put Av = —f. Then A satisfies (3.5) and obviously is a linear operator V —> V*. Using (3.3) we get ,T , \Av\v

\a(v,w)\ , , sup , < j\v\v, v G V, ™ev\{o} \w\v which proves boundedness of A as an operator V —> V* and ||^4||£(v,y) < 7If we define aK(t;, w) = a(u,w) + K(V,W), v,w,e V, then we see (with M 0 being the embedding constant for V C\ H) that =

|aK(w,tu)| < ( 7 + |K|MO)|V|V/|W|V,

(3.6)

ReaK(i>,u) > ^|w|y,

V,W

£ V,

v G V.

Therefore the sesquilinear form aK(-, •) defines an operator AK : V —• V*. The definition of aK implies that (—-AK)(iu) = (—A)(W) + K{V,W), W GV. Therefore we have AK = A — KI. Using (3.4) we obtain the estimate \AKv\v*\v\v

> \{AKv)(v)\ = \aK{v,v)\ > ReaK{v,v)

> 6\v\l,

VGV,

which implies that AK is injective and H^Ml^rangeA^v) - V^- I*1 order to prove surjectivity of AK we choose / G V*. Then, according to the LaxMilgram theorem (Theorem 3.2), there exists a unique v GV such that f(w) =

3.1. Dissipative operators and sesquilinear

forms

93

aK(v,w) = -(AKv)(w), w G V. This proves AK(~v) = / . Thus H A " 1 ^ ^ . ^ ) < 1/5 and, observing |u>|y. < MQ\W\V, W G V, also H A " 1 ! ! ^ . ^ . ) < M$/6, which in particular implies K G p(A). b) Let Av G H. Then by the identification H* = H and the fact that the embedding H Q V* means restriction of functionals on H to V we obtain (3.7)

a{v,w) — -{Av){w)

= -{Av,w),

w G V,

and, consequently, |a(v,«;)| < \\Av\\ \\w\\, w £V. If on the other hand |(Av)(u>)| = \a(v,w)\ < a||ty||, w G V, for some a > 0, then, by density of V in H, Av is the restriction to V of a bounded antilinear functional h G H* = H. This implies Av = h G H. Thus the characterization of dom A given in the theorem is verified. It is easy to see that the part AK of AK in H is given by AK = A — KI and that domA* = dom A. Since H is dense in V*, A~x is continuous and dom A — A^H, we see that dom A is dense in V with respect to the topology of V and consequently also with respect to the topology of H. Finally, by density of V in H, dom A is dense in H. Obviously AK is injective with range AK = H. Moreover, we have (using (3.4)) for v G dom A | | A ^ | | H | > \(AKv)(v)\ = \aK(v,v)\ > ReaK(v,v)

> S\v\l >

~^\\vf,

which proves HA^ftH < (Mg/6)\\h\\, h<= H, i.e., K G p(A). Dissipativity of AK = A — KI follows from Re(AKv,v)

= -ReaK(v,v)

< -S\v\y

< -—^ \\v\\2 < 0,

» e dom A.

If (3.4) is satisfied for some K £ R , then it is also true for some K0 > 0. This implies that range AKQ — H. This and dissipativity of AK show that A is m-dissipative (see Definition 1.11, b)). • The final part of the proof for Theorem 3.4 shows that in fact the operator A — (K — 8/MQ)I is dissipative. 3.5. Corollary. The operator A defined in Theorem 3.4, b), is the infinitesimal generator of a Co-semigroup S(-) G G(l,n). Proof. The result follows immediately from the Lumer-Phillips theorem and Theorem 3.4, b). • The following estimates for the resolvent of A resp. A will be important in the next section. For 6 G (0,7r/2] and K G R the sector T,QtK is denned by (see

Chapter 3. Analytic Semigroups

94

Figure 3.1) S e , K = { A e C | A ^ K , | a r g ( A - / < ) | < - + 0}. ~2 Instead of £# o we shall write T,g.

V

V*

se

/Figure 3.1

Figure 3.2

For later use we observe that (see Figure 3.2) (3.8)

|ImA| > |A|cos0

for A e E e , ReA < 0.

3.6. Theorem. Let a(-, •) be V-coercive. Then there exist a 9 £ (0,7r/2] with He,K U {«} C p(A) n /J(J4) and, /or any e G (0,0), positive constants 7$ = 7i(e), i = 1,2,3, swc/i £/ia£ /or A G £ £ , K , A ^ K , (3.9)

||(i-A/)-1||z;(y,!V)<71(e),

(3.10)

IKi-A/m^.,^

(3.11)

IKi-A/)- 1 !!^.^.)^

(3.12)

IKA-A/)-1!!^^)^

(3.13)

IKA-A/)-1!!^,^^

72(e) |A-K|V2'

73(e) |A-K|' 73(e)

|A-«|' 72(e)

IA-

1/2-

Proof. It is sufficient to consider the case K — 0. Otherwise we replace a(-, •) by aK(-,•) and accordingly A resp. A by A - KI resp. A- KI (compare (3.6)). We first establish the estimates for ReA > 0, A ^ 0. a) For u G V we set / = (A - \I)u G V*. Then, for any v G V, (3.14)

/(u) = - a ( u , w ) - A ( u , t ; ) .

3.1. Dissipative

operators and sesquilinear forms

95

For u = v and Re A > 0 we obtain \f\v\u\v

> |/(w)| > - R e / ( u ) = Rea(u,u)

which implies \(A — XI)u\v (3.15)

+

\\u\\2ReX>6\u\2/,

> <S|«|v, u £ V, or, equivalently,

WiA-Xir'Wav^v)^],

ReA>0, A ^ 0.

b) Formula (3.14) for v = u gives (3.16)

|A| \\uf < | / H | + |a(u,«)| < \f\vMv

+ l\u\2v.

This and (3.15) imply |A|H2<72W., where 7I — (1 + j/6)/6.

The last inequality is equivalent to

Mi-AJ)u|v.

^IAHIUII,

«6V,

which proves (3.17)

||(A-AJ)-l£(v.,ff)<^_,

R e A > 0 , A ^ 0.

c) Equations (3.14) and (3.15) imply |A||(u,v)| < | / | v M v + 7 M v M v
v e V,

-1

where 73 := 1 + j/S. For u = (A — A I ) / considered as an element in V* we have u(v) = (u, v), v € V. Therefore the last estimate gives

\((A-\I)-1f)(v)\
| | ( i - Xir'Wavy)

V€V,

< (73/|A|)|/|y*> which implies < 7^,

ReA > 0, A ^ 0.

d) If u G dom A, then f = (A- XI)u e H and f(v) = (/, v), v € V. For v = u we obtain, for Re A > 0, A ^ 0, 11/11 l|«|| > | / ( « ) | = | ( ( 4 - A J ) u ) ( « ) | = \a(u,u) + X{u,u)\ > Rea(u,u) + ||u|| 2 ReA > 6\u\2v. This and (3.16) imply

|A| \\u\\2 < |/(«)| + |a(«,«)| < 11/11 ||U|| + 7 H 2 , < 73II/II H , resp. ||(4 - A/)" 1 /!! < (73/IADH/ll, i.e., we have established (3.20)

\\{A-)J)-1\\c(H,H)^l[ty

ReA>0, A^0.

96

Chapter 3. Analytic

Semigroups

e) Inequalities (3.19) and (3.20) imply, for u G domA,

s\u\l< H/ll N| 0, A ^ 0, (3.21)

IK^-AJ)-

1

!^)^-^)

In a second step we establish the estimates (3.9) - (3.14) for Re A < 0, A in sectors E £ with e G {0,9) for some 0 G (0, TT/2]. We first prove a result in a more general setting (for later use) than needed here: 3.7. Lemma. Let A be a densely defined operator on a Banach space X and assume that { A | R e A > 0 , A ^ 0 } C p{A) and that there exists a constant 7 > 0 with (3.22)

\\{A-\I)-l\\C{x,x)
R e A > 0 , A ^ 0.

Then there exist a constant 9 G (0,7r/2] with T,g C p{A) and, for any e 6 (0,9), a constant 7(e) > 0 with (3.23)

11(^4-AJ)-1!!^,^^^,

A G E e , A ^ 0.

Proof. For A — /x + IT we have {A - XI)'1 (3.24)

= {A- irl)-1

{I ~ n{A -

= {A- iTl)-l^pi{A

-

irl)-1)"1

iriyi,

j=o

provided |/z| \\{A - irl)

(3.25)

1

\\c(x,x) < 1- We choose c G (0,1). Then T\

1

Ul-KA-*I)-rX[x^)<-^
for \fi\ < C | T | / 7 . If we define e = arctan(c/7), then \H\/\T\ = tan(|argA| 7r/2) < tane = c/7 and consequently (3.24) is true for A G S e , Re A < 0. Since c G (0,1) was arbitrary, this proves also Y,g C p{A), where 9 = a r c t a n l / 7 . Moreover, (3.8), (3.22), (3.24) and (3.25) imply \\{A- Xir'Wc^^

< -^7^— < jz ^ 7^7, A G S £ , R e A < 0 , \T\ 1 - c (1 - cjcose |A| which together with (3.22) proves the result.

•

3.2. Analytic semigroups

97

Proof of Theorem 3.6, continued. The estimate (3.12) follows immediately if we apply Lemma 3.7 using (3.20). In order to prove (3.13) we use (3.24), (3.25) for X = H and (3.21), (3.8). We get

\\{A-\I)-1\\C{H,V)

< <

MA-iTiy'Wc^Wil-^A-iTiry'l ^

1

<

C(H,H)

72

IV2 1 - C - (1 - c)Vcose |A|V2 for A G %, ReA < 0. This together with (3.21) proves (3.13). The proof for (3.9) - (3.11) is completely analogous. One has to use the estimate

ll(/^(i-^rr'll A V ,., l ,,< F Ji R < T ^ for \fi\ < c|rI/73 and the estimates (3.15), (3.17), (3.18).

•

The most important properties of the operators A and A are the estimates (3.11) and (3.12), respectively. This gives rise to the following definition. 3.8. Definition. A densely defined operator A on a Banach space X is called a sectorial operator on X if there exist numbers 6 G (0,7r/2] and K G R such that (i) E 9)K C p(A) and (ii) for any e G (0,6) there exists a constant 7 = 7(e) > 0 with (3.26)

IKA-A/)-

1

!!^,^-^-,

A€S£,

M

A^.

\A - K\

Lemma 3.7 implies that we only need to establish the estimate (3.26) in a half-plane, i.e., for e = 0, in order to verify that an operator is sectorial. A straightforward consequence of Theorem 3.6 is the following result: 3.9. Proposition. Under the assumptions of Theorem 3.4 the operators A and A defined in that theorem are sectorial operators on V* resp. on H. In addition we have K G p(A) n p{A) (K the constant in (3.4)). 3.2.

Analytic semigroups

Throughout this section X will be a complex Banach space. For a bounded linear operator A on X the mapping z -> eAz, z G C, is analytic on C and satisfies the semigroup property eA(Zl+Z2^ = eAzieAz2, zi,z% G C. In this section we investigate semigroups of bounded linear operators on X which are analytic on some subdomain of C. Furthermore, we deal with the question, when a Co-semigroup can be extended to an analytic semigroup. Since the

Chapter 3. Analytic

98

Semigroups

subdomain C of C, where such a semigroup is defined, has to be an additive subsemigroup of C, there are two possibilities, C = C or C is a cone in the right half plane with vertex at 0. For simplicity we restrict ourselves to symmetric cones around the real axis, Ca := {z G C | z =£ 0 and | arg z\ < a} U {0}, where a € (0,7r/2]. 3.10. Definition. Let either C = Ca with 0 < a < n/2 or C = C and assume that (5(z)) „ is a family of bounded linear operators on X. S(-) is called an analytic semigroup (or holomorphic semigroup) on X if and only if (i) 5(0) = / and 5(zi + z 2 ) = 5(zi)5(z 2 ) for all z 1; z2 G C, (ii) for any x G X and x* e X*, the mapping z-> (x*,S(z)x),

z eC,

o

is analytic in C and (hi) for any x £ X and e G (0,1), lim_ S{z)x = x resp. z->0, 2 6 C e Q

lim S(z)x = x if C = C z

->0

From basic results on analytic vector valued resp. operator valued functions (cf. for instance [H-P]) it follows that condition (ii) is equivalent to o

(ii') for any x € X the mapping z -¥ S(z)x is analytic in C or to o

(ii") the mapping z —> S(z) is analytic in C. The following observations are an immediate consequence of (ii"): 3.11. Proposition. Let {S(z))zeC be an analytic semigroup onX. a) The mapping z —» S(z) has continuous derivatives of arbitrary order in C with respect to the uniform operator topology. b) IfC — C then there exists a bounded linear operator A on X such that S(z) = eAz,

zeC.

Proof. Statement a) is obvious from (ii"). If C = C then z —> S(z) is an analytic function C -> C(X). We set A = S'(0) G C(X). Then we have S'(z) = \imw^0(l/w)(S(z + w)- S(z)) = S(z)limw^0{l/w)(S{w) - I) = S{z)A and consequently 5"(0) = Yimz^0{\ / z){S'{z) - A) = limz^0(l/z){S(z)-I)A = A2. By induction we see that 5 ^ ( 0 ) = A>, j = 0 , 1 , . . . . The Taylor expansion of S(z) gives the result. • Of course, the restriction of an analytic semigroup on X to R is a Cosemigroup on X. Next we want to address the question when a Co-semigroup

3.2. Analytic semigroups

99

on X can be extended to an analytic semigroup on X. Necessarily such a semigroup has to have continuous derivatives of arbitrary order on t > 0 with respect to the uniform operator topology. The next proposition characterizes this property. 3.12. Proposition. Let S(-) be a Co-semigroup on X with infinitesimal generator A. Then the following statements are equivalent: (i) S{t)X C domA for all t > 0. (ii) The mapping t —>• S(t), t > 0, is differentiable on t > 0. / / (i) or, equivalently, (ii) is true, then t ->• S(t) has derivatives of arbitrary order for t > 0 and S{k\t)=AkS(t),

(3.27)

t > 0, Jfe = 1,2

Proof, a) Assume that (i) is true. We first prove that (i) implies S(t)X cdomAh,

(3.28)

t > 0, k = 1,2,... .

Fix k € N and assume that, for any x e X and i = 1 , . . . , k - 1, we have S(t)x € domA 1 , t > 0. Then using also Proposition 2.11, b), we get Ak~1S(t)x

= S'(i/fc)(A5(i/fc))fc"1a; G domA,

t > 0,

fe

i.e., S(t)x G domA , t > 0 , i £ l Since Afc, fc = 1,2,..., is closed (cf. Proposition 2.11, c)), k A S(t), t > 0, k = 1, 2 , . . . , are bounded linear operators on X graph theorem. From (3.28) and Proposition 2.11, b), we see to > 0 and i f X , the mapping t -> (dk/dtk)S(t)x is continuous £s{t)x

= £-kS(t0

+ r)x =

= AkS(r)S(t0)x

the operators by the closed that, for any for t > t 0 and

£-S(r)S(to)x

= AkS{t)x,

t > t0, k = 1,2,... .

Integration yields rt+h r-t-re

AkS(r)xdT

/ V»)

,t+h = /

S(T-t0)AkS(t0)xdT

for any t > to, x e X and \h\ < t — to- This implies the estimate \(Ak-1S(t

+ h)-Ak-1S(t))x\<\h\\x\\\AkS(t0)\\

sup 0
and, consequently, ||A fc - 1 5(t + h) - Ak-lS{t)\\

= 0{\h\)

ash^O

||5(r)||

Chapter 3. Analytic

100

Semigroups

for t > to and k = 1,2,... . This proves that the mappings t —> AkS(t) are continuous for t > 0 with respect to the uniform operator topology. Therefore (3.29) implies A^Stf

+ h)- A"-1S(t)

ft+h = / AkS{r) dr

for k = 1,2,..., t > 0 and |ft| < £, i.e., the mappings t ->• AkS(t), k = 0 , 1 , . . . , are continuously differentiable on t > 0 with respect to the uniform operator topology and (d/dt)Ak-1S(t) = AkS(t), t > 0, k = 1,2,... . For fc = 1 this shows that t -> 5(i) is differentiable on t > 0 and S"(t) = AS(t), t > 0. An induction argument completes the proof for (3.27). b) If (ii) is true, then we get, for any x G X and t > 0, lim | (S(t + fc)i - 5(t)a:) = flim | (S(t + h) - 5 ( t ) ) ) x = S'(t)a:, i.e., we have S(t)x & domA.

D

A Co-semigroup 5(-) on X satisfying (i) resp. (ii) of Proposition 3.12 is called a differentiable semigroup on X. Of course, a differentiable semigroup S(-) in general does not have an analytic continuation. As we shall see in the next theorem the behavior of S(-) as t \. 0 is crucial. 3.13. Theorem. Let S(-) be a Co-semigroup on X with infinitesimal generator A. Then S(-) can be extended to an analytic semigroup on X if and only if S(-) has the following two properties: (i) S(t)X C domA for all t > 0 (i.e., S(-) is a differentiable semigroup) and (ii) b := limsupmt\\AS(t)\\ < oo. Moreover, b < 1/e implies that A is bounded and hence the extension of S(-) is given by S(z) = eAz, z G C. For b > 1/e the extension ofS(-) is defined at least on the sector CQ, where 0 < a = arcsin(6e) _1 < 7r/2. If (i) and (ii) are satisfied, then we have, for any x G X, (3.30)

limtkAkS(t)x

= 0,

k = l,2,...

.

Proof, a) We first assume that S(-) has properties (i) and (ii). From Proposition 3.12 we see that t —>• S(i) has derivatives of arbitrary order on t > 0. Hence, for any £Q > 0, Taylor's formula implies the representation (note that

3.2. Analytic semigroups

101

£<*>(*) = 4*S(<), t > 0) n-l fc=0

(3.31) fl»(i) =

r

4rw

(_n

ij. j

/ (* - i-) n _ 1 A n S(T) dr,

t > 0.

t o

For arbitrary K > 0 there exists a J > 0 such that t\\AS(t) || < 6 + K for t G (0, (5]. Using this and also Stirling's formula we obtain the estimate

(3.32)

h.llAks{T)^ - h ) l A S { T / m k * h b + K ) k { k / T ) k <(e{b + K)/t0)k,

t0
provided k is sufficiently large, e.g. k > t/S. This implies the estimate i(b + K) i ||J*»(t)|| < ( ^ - ^ ( t - to))", n > t/S. to Since K > 0 was arbitrary we see that lim.Rn(t)=0

for0
n-*oo

e6

Therefore we get from (3.31) the expansion 00

1 S(t) = £ - ( t - t 0 ) fc A fc S(t 0 ),

t o
fc=0

Consequently this power series converges for \z—
5(z) = J ] - ( z - t o ) f e A f c 5 ( t o ) ,

|z-t0|<^.

fe=0

The family of functions defined by the series (3.33) for all to > 0 gives an analytic continuation of the original semigroup S(t), t > 0, on C = \Jto>0{z G C | \z — to| < to(e&) -1 } U {0}. In case eb < 1 we have to(e6) - 1 > t 0 , which implies C = C. By Proposition 3.11, b), A is bounded and S(z) = eAz, z G C. If e& > 1 then C — Ca with a = arcsin(efc)-1. It remains to prove that (S(z))z c satisfies conditions (i) and (hi) of Definition 3.10. For fixed f > 0 the mappings z -» S(t + z) and z —>• S{t)S(z) are analytic o

in Ca and coincide for z G R, z > 0. This proves S(t + z) = S(t)S(z) for all t > 0 and z G Ca- Then, for fixed z0 G Ca, the mappings z -4 5(z -f zo) and o

z ->• S(z)S(zo) are analytic in CQ and coincide for z G R, z > 0. This proves S(z + z0) = S(z)S(zo) for all z, z0 G Ca.

Chapter 3. Analytic

102

Semigroups

t0/eb t

to

Figure 3.3

Z P

p/ COS2 ea

Figure 3.4

In order to verify property (iii) of Definition 3.10 we set, for 0 < e < 1 and p>0, Cea,p = {z€C€a

| 0 < R e Z < p}

and choose the unique e G (0,1) such that sinea = e(e&)_1. For z £ CeatP we choose t > 0 such that £sin|argz| = \z — t\ (see Figure 3.4). Note that t < p/cos 2 ea for z € Cea>p. The choice oft implies \z — t\ < isineo: = ie(e&) _1 . Using this and the estimate (3.32) (with r = t = to) we get

for k > k0 > p(S cos2 e a ) - 1 . For k < k0 we see from the choice of S that \\AkS(t)\\ < \\AS(t/k0)\\k k

||5((fco -

k

<(b+K) (k0/t)

k)t/k0)\\

SUP 0
\\S(t)\\

and therefore jJ f ||A f c 5(t)|||z-t| f e
fc!

V e 0 /

< \

eo

/

™p 0
0
||5(t)||

3.2. Analytic semigroups

103

We choose K > 0 such that e(b + K,)/b < 1. Then we get from the power series expansion for S(z) 00

I

+ 1^

\\s{z)\\<\\sm^Y^r\\Aks^\\ fc=l

(3.34)

<

sup ||5(0|| (l + ^ E ^ ^ r ) * ) o
For a: G X and 77 > 0 we choose s > 0 such that |x - 5(s)a;| < 77. Since lim2_j.o S(z)S(s)x = limz_».o 5'(2: + s)a; = S(s)x, we can choose 5 > 0 such that \S(z)S(s)x — S(s)x\ < 77 for \z\ < 6, z G C£QjP. Hence we get for these z the estimate \S{z)x -x\<

\\S(z)\\ \S{s)x -x\ + \S(z)S(s)x

- S(s)x\ + \S(s)x - x\

< (M £iP + 2)7?, which proves limz_j.o, zec€a S(z)x = x. b) Assume that S(-) can be extended to an analytic semigroup on X. Then property (i) is true by Proposition 3.12. In order to verify property (ii) we can assume that S(z) is defined on some sector Ca with 0 < a < TT/2. We choose e G (0,1) and set p = 1 + sinea. By continuity of S(-) there exists a constant Mo > 1 such that \\S{z)\\<M0,

zeCea,

\z\
Using Cauchy's integral formula we obtain

A*S(t) = SW(t) = ±-.

l {c_ t)k+1S(C)dt,

J

0
|£—£|=£ sinea

and consequently Mo_ 1 < —jr--^, 0 < * < 1 , fe = l , 2 0 sin'" sin ea ea 1tK which in particular proves (ii). This implies also that for any k > 1 there exists a constant 7fc > 0 such that ||rjfc^4fc5(t)|| < 7fc for 0 < £ < 1. For x G domA k we get (see also Proposition 2.11, b)) \AkS(t)\\

\imtkAkS(t)x «|0

Since domAk ment.

= limtkS(t)Akx

= 0.

UO

is dense in X, the proof is finished by a standard density argu•

104

Chapter 3. Analytic

Semigroups

3.14. Corollary. Under the assumptions of Theorem 3.13 the derivatives of S(z) are given by S(k){z)

= AkS(z),

z£Ca, fc = l , 2 , . . . .

Proof. From (3.33) we see that oo sU)

1

= E !?(* - t0)kAk+JS(t0),

W

k=o

\z - to| <

t0(eb)-\

'

By closedness of Aj we have CO

^

OO

- ( z - io) f c A f c +^( i o ) = ^ £

£ fc=0

'

1

- ( 2 - io)fc^5(i0)

fc=0 j

= A S(z),

\z-to\Ktoieb)-1.

Since io > 0 is arbitrary, this proves the result.

•

Let (S(z))zeC , 0 < a < n/2, be an analytic semigroup on X. For any 6 G {—a, a) the operators Se(t) = S(tel9), t > 0, constitute a Co-semigroup. Let Ag denote the infinitesimal generator of Sg(-). 3.15. Lemma. For any 0 G (—a, a) we have dom Ag = dom^o and Agx = el6Aox for x G dom^lo- Moreover, for any e G (0,1) we have (3.35)

lim_

- (S(z)x - x) = A0x,

x£domA0.

z-t-0, z g C 6 Q Z

Proof. For z G Cea we set t = \z\. Obviously z ->• 0, z G Cea, implies t 4- 0. Then we get for x G dom J4 0 - (S(z)x -x)~

A0x = ^—^ (—!— (S(z)x - S(t)x) -

A0x)

+ t(l(S(t)x-x)-A0x)=:Z—±I1

+

il2.

Obviously we have lim^ 0 I2 = 0. For I\ we get (see also Corollary 3.14 for k = 1 and A = A0) \h\

\z-t\

[(S(C)~l)A0xd{

<

sup _ ICI
\(S(Q-l)Aox\,

where i, z denotes the line segment from t to z. In view of property (iii) in Definition 3.10 this proves lim 2 ^ 0 ze^ ij = 0. This finishes the proof of (3.35).

3.2. Analytic semigroups

105

Using (3.35) we get , for x € dom An, 1 eie - (Se(t)x - x) = —r (S{tel9)x - x) -> e'eA0x ast|0, t re 1 i.e., we have x G domA0 and A^a; = el9A0x. On the other hand we have, for x G dom Ag (3.36)

Aoz = lim i-S9(t)x *4.o at = eielim

= ei0 lim 5'(te i e )x tio

A0S{tei6)x,

where we have used Theorem 2.9 for Se(-) and Corollary 3.14 for S(-). Frcrtn (3.36) we see that limt_|_o AoS(tel6)x = e~x9Aox. Closedness of AQ implies x G dom AQ and AQX = e~l6Aex. D" In view of Lemma 3.15 we call AQ the infinitesimal generator of the analytic semigroup (S(*)) z e C a .

Figure 3.6

Figure 3.5

The Hille-Yosida theorem characterizes those operators which are infinitesimal generators of Co-semigroups. In the following we characterize infinitesimal generators of analytic semigroups. 3.16. Proposition. Let (S(z)) c , 0 < a < n/2, be an analytic semigroup with infinitesimal generator A. Then for any e G (0, a) there exist constants Ke G R and Me > 1 such that S a _ e , K e \ K } C p{A)

106

Chapter 3. Analytic

Semigroups

(see Figure 3.5) and \\(\I - A)-1]] < - ^ - , |A -

AeEa_e,Ks,A^,

Ke\

In particular, A is a sectorial operator. Proof. We choose e <E (0,a). Since Sa-€(-) and 5_ Q+£ (-) are Co-semigroups, there exist constants M£ > 1 and a ) e e M with \\Sa-e(t)\\

< Mte^1

||S_ Q + £ (t)|| < M£ew'*,

and

t > 0.

For A G C with | arg(A - w£)| < (n - e)/2 we have |A — u;£| sine/2 < ReA — u>e. This and the Hille-Yosida theorem imply ||(A7 - A , - , ) " 1 ! ! <

|A

_J^£sine/2,

I arg(A - w £ )| < (TT - e)/2, A ^ c*.

Using A a _ e = e ^ " - 6 ) ^ (see Lemma 3.15) we obtain for \i = Ae _ i ( a _ e ) the estimate (3.37)

W^I-Ar'WK*?\ , //— w£e ^ a ^ | sin e/2

jxe/C, / x ^ e - ^ ) ,

where /C={

M

eC|-a-| + |<arg(/,-a;

£

e^-))<|-a +|}.

We define K£ = w£ cos(a — e) (l + tan(a - e) tan(a - e/2)) = u)e cos e/2 / cos(a - e/2) (compare Figure 3.7). Then the cone C_ defined by C_ = {n e C | - Q - | + y < arg(/i - «£) < 0} is a subset of the cone K. Since ute~%^a~^ exists a constant C£ such that

-,—^~Ke}

is bounded away from C_, there

,,
HGC.

This and (3.37) imply

Analogously, starting with the estimate for the resolvent of J 4 _ Q + £ we obtain the same estimate for \x with 0 < a r g ( / i - « e ) < a + 7r/2-3e/2. Altogether we have £a-3£/2,K£ \ { K J

C

Z9^)

3.2. Analytic

semigroups

107

Figure 3.7

and

Kfil-AY

Me

<

/i G £ Q _ 3 e / 2 , K e , M 7^ Ke>

with an appropriately defined constant Me > 1.

•

In order to state a result converse to Proposition 3.16 we need some notation. For any S > 0 and 7 G (0, TT/2) we define the contour Ts^ by

(3-38)

r,,7 = r+ 7 ur,; 7 ur° 7 ,

where (see Figure 3.8) (3.39)

6,-y

r°

{A G C I IA| > S, arg A = ±(TT/2 + 7)}, {AGC||A|=<S,

<S,7

|argA|<7r/2 + 7 } .

The orientation of this contour is such that Im A is increasing along Tf Furthermore we set r,5 i7 + {K} = {A + K | A G r,j ) 7 }. The following lemma will be useful: 3.17. Lemma. Let 0<9<e
and

TT/2 and z, A G C with — + a — 9 < arg A < —

a +9

108

Chapter 3. Analytic

Semigroups

be given. Then we have ReXz < -\X\ \z\ sin(e - 9) < 0. Proof. By assumption on A and z we get — + e-9

< arg \z = arg A + arg z < —— (e - 9).

This implies cos(arg Xz) < cos(7r/2 + e - 9) = - sin(e - 9).

a

Figure 3.8

Figure 3.9

3.18. Proposition. Let A be a sectorial operator on X, i.e., domA = X and there exist constants K 6 M., a G (0,7r/2] and for any e G (0, a) a constant Mt > 1 such that (see Figure 3.6) £a, K C p(A) and (3.40)

KAJ-A)-1!! <

M( \\-K

,

A G Lja—(K,

A f K.

T/ien A is the infinitesimal generator of a Co-semigroup which can be extended to an analytic semigroup (S{z))z . Moreover, for any z G Ca, z ^ 0, and k — 0 , 1 , 2 , . . . the derivative S^(z) (3.41)

SW(z) = AkS(z)

is given by

= -^ / 2i"

XkeXz{XI - A)~ldX,

./IY^-KK}

3.2. Analytic semigroups

109

where 5 > 0 and0 < 0 <

a—\argz\•

Proof. It suffices to consider the case K = 0. Otherwise we replace A by AK = A — KI which satisfies the assumptions with K = 0. If AK generates a Co-semigroup SK(-) which can be extended analytically into the sector Ca, then A itself generates S(t) — e Kt 5 K (t) which obviously can also be extended analytically into Ca. For z e C a , z ^ 0, we set e = a — | argz| and define (3.42)

S{z) = —

eXz{XI - A)~xdX,

f

where (5 > 0 and 0 € (0, e). Then, for A G r £ Q _ e , Lemma 3.17 and (3.40) (with K — 0) imply ||e A *(A7- A)" 1 1| < e ^ ^ ^ f

< e-W W ' ^ - ' ) ^ ,

I'M

A G T±a_e.

1^1

This estimate and continuity of A —> eXz(XI — A)"1 on T*Q_^ prove that the integral along Ffa_e exists. Existence of the integral along T" a_e is trivial. An application of Cauchy's theorem implies that Ts,a-9 in (3.42) can be replaced by any Fs',a-ff' with 5' > 0 and 0' G (0, e). This shows also that S(z) is defined on Ca \ {0} by (3.42). For z = 0 we set 5(0) = 7. We first prove that S(z), z € Ca, is a semigroup on X. For 2 , i « 6 C a \ {0} we set e = min(a - | a r g z | , a - | argK;|) and choose 0 < 6\ < 02 < e and 0 < Si < S2- Then we have with Tj = Tsua-9i, i = 1,2, (see Figure 3.9) S(z)S(«>) = T ^ T T /

/

e ^ e ^ A J - A)-\nI

= [2-Kiy 7T^ J I J I Tl

= 7^—xo/

r2

e^iXI-A)'1

2

(27rt) 7 ri

-

A^dfidX

e'^-l—UXI-Ar'-^I-AT^dndX fj, - X /

-dftdX

7r2M-A

" 7(2m) 5 ^ 22 7/ e * " 0 ^ / - ^ ) - 1 y/ r2

ri

/x - -^rdXd», A

where we have used the resolvent identity, Fubini's theorem and the fact that /i ^ A for A G Ti, /x G rY Simple computations show that 1 f e^w (3.43) — / r dji = eAtu for A G Tj 27n J r 2 /i - A and (3.44)

- ^ f -^dX 2™ y F l n - X

=0

forMGr2.

110

Chapter 3. Analytic

Semigroups

Therefore we get (observe | arg(z + w) \ < a — e) S{z)S(w) = ^-.

2-KI

f ex(-z+w\XI-A)-1d\ JTi

= S(z + w),

which establishes the semigroup property for S(z), z G Ca. Standard results on functions of complex variables show that, for arbitrary x £ X and x* G X*, eXz((XI - A)~lx, x*) d\

z -> (S{z)x, x*) = -^-. [

is analytic on Ca-c for any e G (0, a), where S > 0 and 6 G (0, e) are fixed. This proves that property (ii) of Definition 3.10 is satisfied on Ca. Before we prove property (iii) of Definition 3.10 we show that for any e G (0,a) there exists a constant Ke > 1 such that ||5'(2)|| < Ke for z G C a _ £ . We fix 5 > 0 and 6 G (0, e). Then we have, for z G C Q _ € , z^Q, S(z) = —

f

eXz{\I-A)-1d\

= ^~

Let z = \z\eiTl. For /i G Tja_g we have // = aei^/2+a-e\ estimates (using (3.40) and Lemma 3.17) give m

K z

Jvin_a

J

\\

\z\

2w

eXz{\I-A)-xd\

(

6 < a < oo. Simple

Js

o

The same estimate holds for the integral along Fs a_9. Along r%a_0 we obtain (H = SeiT, \T\ < TT/2 + a - 9) the bound eScoB<~T+^M9dT =: K0.

— / Altogether we have shown that (3.45)

\\S(z)\\<2K1

+ K0,

z £CQ_,

For x G dom A and z G Ca_<:, z ^ 0, we obtain (with 5 > 0 and # G (0, e))

S(z)x-x

= -^-. f 1

/"

e^dxi-A^-^-AxdX pAz

= — / 2Wr,„_„ A

-rr-iXI-A^AxdX.

3.2. Analytic semigroups Here we have used

1

(3.46)

eXz

I

d\ = l.

For each A e r^ ) a _e we have 0 Az

i

— (A/ - A)'1 Ax -> y (AJ - A ) _ 1 A r

a s z - > 0 , z £ CQ_£

and the estimates M9l |A|

— (XI-A)-1

Ax

— {XI-A)'1

Ax <

Aer±Q,


Ax\, AerL_ e .

A

Therefore by the dominated convergence theorem we have

lim

(3.47)

Here

(S(z)x-x)=—

v ec„_. z->o, zec Q

I^Q-S^,

f

2mJT

X

^(XI-A^AxdX

= lim - i r / |(A/-^)_Ma:dA *->«> 2TT* ./r,,„_ SiR A R> 6, denotes the contour depicted in Figure 3.10.

r,5,a-0,H

Figuree 3.10

112

Chapter 3. Analytic

Semigroups

For arbitrary x G X we choose a sequence (xn) C dom A with xn —> x as n —>• oo. Then, for z £ C a - e , \S(z)x -x\<

\\S(z)\\ \x - xn\ + \S(z)xn - xn\ + \x-

xn\.

Using (3.45) and (3.47) we obtain lim_ \S{z)x - x\ = 0, z^-o, zeca-e

xeX.

We want to prove next that the infinitesimal generator of (S(z))zeca indeed is A. Let B denote the infinitesimal generator of S(-) and assume that we have established the following result: For any x £ domA we have (3.48)

^S(t)x = S(t)Ax, t > 0. at Of course this would be trivial, if we already knew that A is the infinitesimal generator of S(-). Equation (3.48) implies that 1 1 fh lim —v (S(h)x — x) = lim — / S(s)Axds Ho h ' ' hio h J0

= Ax,

i.e., x G dom B and Bx = Ax. Thus B is an extension of A. Obviously we have p(A)Hp(B) ^ 0, which shows that for any x G domB there exists a y G domA such that (A0J - B)x — (X0I - A)y = (X0I - B)y. By injectivity of A 0 / - B we get x — y £ domvl, i.e., A = B. In order to prove (3.48) we shall first verify the formula (3.41) for the derivatives of S(z). For z G Ca, z ^ 0, choose S > 0 and 0 < 6 < a — | argz|. Then Lemma 3.17 with e = a — \ arg z\ and the estimate (3.40) imply (for some constant M > 1) ||A*e A *(AJ-A) - 1 !! < M | A | f c - 1 e - | A | | z | s i n ( Q - | a r g z l - ^ for A G Tfa_e. This estimate proves that the integrals J r \keXz(XI — A)~ld\ exist for all k = 0 , 1 , 2 , . . . . Formula (3.41) is trivially true for k = 0 by definition of S(z) in (3.42). Assume now that (3.41) is already established for k. Then we have

hs^(z h

+ h)-S^(z))=^-:

\{eXh~l)\keXz{\I-A)-ld\.

[ l-Kl JYs
For any A G Ts,a-6 the integrand converges to Xk+1eXz(XI — A)~l as h —• 0. Moreover, we have the estimate (cf. Lemma 3.17 and (3.40)) i(eA,l-l)AfceA2(A/-J4)-1 h

< |A| e l A H' l l|A| f c e- | A | | z | s i n ( a - | a r s 2 l- 9 ) — |A| Isz\-0)) M|A|fckJ\\(\h\-\z\sm(a-\i e

3.2. Analytic semigroups

113

Therefore we get by the dominated convergence theorem S<*+1>(z) = — f Xk+1eXz(XI 2 ™ Jr,,a-,

- A)"1^.

For z = t > 0 and a; G dom A we obtain (3.49)

S'(t)x = ^ - f

Xe^iXI-A^xdX.

Observing A (A/ - A ) - 1 x — x-\-{XI-A)~lAx 0 we obtain from (3.49) and (3.42) that -rS{t)x

= S'(t)x = S(t)Ax,

for x G dom A and/ r5Q _e At 2;eL\ =

x £ dom A, t > 0,

i.e., (3.48) holds. Now we know that A is the infinitesimal generator of S(-), so that S^(z) = AS(k~l\z) = AkS(z), zeCa,z^0 (see Corollary 3.14). Thus the proposition is completely proved in case K = 0. For general K everything follows by standard arguments (considering A — KI instead of A) with the exception of formula (3.41). Assume now K ^ 0 and let SK(-) denote the semigroup generated by AK = A - KI. We have SK(z) = e~KZS(z), z G Ca. For the derivatives of SK(-) we have the representations SW

J-/"

XkeXz(XI-AK)~1dX

= ^— /

(/* - *)k^z{»i

=

- A)~ld^

fc

= o, l , 2 , . . . .

For k = 0 we get S(z) = eKZSK(z) = - L /

e"*(MJ - A ) - 1 ^ .

Assume now that (3.41) is already established for j = 0 , . . . , k — 1. Then we get z ( k) S K (z)

= ~

f

= ^~.i 'Ts ,-g 2j,

^e^ifil

- A)'1 dpi

M l k A *(/rf Ar d» + £ ( ) (-Kys^\z) + {K} j

ir, aiC_. + {4

= 1

t^\3J

114

Chapter 3. Analytic

Semigroups

or, equivalently, (3.50)

^- f

^/"(fil-A)-^

= e^W(z)-tP)(-^SM(4 j=iVJ'

On the other hand we get from SK(z) = e~KZS{z) the equation eKZSJtHz) = S ( % ) + J2 Using this and (3.50) we obtain (3.41) for k.

(k){-K)SS{k-j)(z). •

Combining Propositions 3.16 and 3.18 we obtain the main theorem for analytic semigroups: 3.19. Theorem. A linear operator A on X is the infinitesimal generator of an analytic semigroup on X if and only if it is sectorial.

CHAPTER 4

A p p r o x i m a t i o n of Co-Semigroups The goal of this chapter is to provide a rather complete discussion of approximation results for linear Co-semigroups. The only omission is that no results on time discretizations are given. See however the presentation of Chernoff's theorem in Section 10.2 for general evolution problems. The fundamental approximation result for linear Co-semigroups is the Trotter-Kato theorem (see [Tr]) which is presented in Section 4.1 in several versions. Our approach is the one given in [I-K-S] which is based on the one by T. Kato (see [Kal]) resp. the one given in [Ki] for the version which assures also existence of the limiting semigroup. Motivated by the fact that Co-semigroups frequently appear as solution semigroups for abstract Cauchy problems stated in variational form we present in Section 4.3 also variational formulations of the TrotterKato theorem (see [I-K2] and [I-K-S]). Approximation of nonhomogeneous problems is based on a product space formulation (see for instance [Bb3] resp. [Daf-Sl]) which reduces the problem to a homogeneous problem (Section 4.2). In Section 4.4 we show that we can get sharp rate estimates in case of almost selfadjoint parabolic problems using Nitsche's trick.

4.1.

The Trotter-Kato theorem

In this section we consider approximation of Co-semigroups and present the basic versions of the Trotter-Kato approximation theorem. Let Z, Xn, n = 1,2,..., be Banach spaces with norm | • | resp. | • \n and X be a closed linear subspace of Z. We assume that for every n = 1,2,... there exist bounded linear operators Pn : Z —> Xn and En : Xn —> Z satisfying the following conditions: (al)

There exist positive constants Mi, M 2 such that \\Pn\\ < Mx

and

\\En\\ < M2 115

for all n e N.

Chapter 4. Approximation

116

(a2)

of

Co-Semigroups

For all x £ X we have lim \EnPnx - x\ = 0. Tl—fOO

(a3)

PnEn = In (= identity operator on Xn), n = 1,2,... .

As we shall see, (a2) is a consequence of the assumptions made in the Trotter-Kato theorem. However, since in practice one usually starts with the definition of the spaces Xn and the operators P„, En, one has to guarantee that assumptions (al) - (a3) are satisfied. We can equivalently use a setup where we consider subspaces of Z instead of the spaces Xn. Set Zn = rangeE n

and 7r„ = EnPn,

n = 1,2,... .

The subspaces Zn C Z are endowed with the norm induced from Z. It is easy to see that the Zn are closed subspaces of Z and that the 7r„ are projections Z —» Zn (i.e., we have -K\ = 7r„ and rangeir n = Zn). Assumption (al) implies (al)

K||<M,

n = l,2,...,

for some positive constant M, whereas assumption (a2) implies (52)

limn^oo irnz = z

for all z £ X.

If (52) is true for all z £ Z, then (51) follows by the uniform boundedness principle. In general the Zn will not be subspaces of X. There are situations where it is of advantage to consider the approximations in subspaces of a larger space Z than the state space X for the problem to be approximated. For an example see [I-K2], where an approximation scheme for Stokes' equation is investigated. Conversely let Zn, n = 1,2,..., be a sequence of subspaces of Z with projections nn : Z ~> Zn and assume that (al) and (52) are satisfied. We set Xn = Zn, Pn = -Kn and define En to be the canonical injection Zn -> Z, Then (al) - (a3) are satisfied. The most frequent situation, where the setting on which assumptions (al) - (a3) are based occurs, is given when we start with a sequence of finite dimensional subspaces Zn C Z, dimZ„ = kn, and corresponding projections 7r„ : Z -> Zn such that (al) and (52) hold. For each subspace we choose a basis 2™,..., z% and define the linear isomorphisms pn : Zn —> Xn = Rkn (or Cfcn) by pnz — (ai,..., afcn)T for z — X]j=i o-iZ^ £ Zn. The canonical injection Zn ->• Z is denoted by i„ and the norm | • |„ on Xn is given by \x\n — [p^x].

4.1. The Trotter-Kato

theorem

117

If we define the mappings Pn : Z -* Xn and En : X„ -*• Z by -PnZ = PnTTn-2,

Z £ Z,

then assumptions (al) - (a3) are satisfied. The concept of convergence used in connection with Co-semigroups is that of uniform convergence on bounded i-mtervals. This has some important consequences as can be seen from the following lemma: 4 . 1 . L e m m a . Assume that conditions (al) - (a3) hold and let Sn(-), 1,2,..., be Co-semigroups on Xn with infinitesimal generators An. If S0{t)x := lim

n =

EnSn{t)Pnx

exists in X for all x € X and t > 0 and is uniform for t in bounded intervals, then the operators So(t) constitute a Co-semigroup on X. Moreover, there exist constants M > 1 and u £ R such that Sn(-)<EG(M,w), Let

AQ

n = l,2,...,

and S0{-) £

G{M,u).

denote the infinitesimal generator of SQ{-)- Then we have lim \En(XIn - A^PnX

- (XI - Ao^xl

=0

for all x & X and A with Re A > iv. Proof. It is clear that t -» So(t)x is a continuous function [0, oo) —> X for all x £ X. From (a2) we get 5o(0)a; = lim EnSn(0)Pnx

= lim EnPnx

n—voo

= x,

x £ X.

n—>oo

We define the operators Sn : X ->• C(0,1; X) by (Snx) (t) = EnSn(t)Pnx,

xeX,

0 < t < 1.

Uniform convergence of Sn(t)x on bounded intervals implies that for any x £ X there exists a constant M(x) such that \S„x\C(0,l;X)<M(x),

71 = 1,2, . . . .

By the uniform boundedness principle there exists a constant M > 1 such that sup max. \EnSn{t)Pnx\

= \\Sn\\ < M,

n = l,2,... .

|Z|<1°<*<1

This implies \\EnSn(t)Pn\\ < M for all t £ [0,1] and n = 1,2,... . Consequently, we have for any xn £ Xn < M1MM2\x„\n, where we have also used (al) and (a3). This proves \\Sn(t))\<M,

0
0 < t < 1,

118

Chapter 4. Approximation of Co-Semigroups

with M = MiMM2- Any t > 0 can be written as t = k + s with s G [0,1) and k G No- Therefore we have \\Sn{t)\\

\\Sn{l)\\k\\Sn{s)\\

= \\Sn{k + S)\\ =

<MMl

= MetlogM,

i > 0 , n = l,2,... .

In order to prove that SQ(-) has the semigroup property we observe that, for all x € X and t, s > 0, lim EnSn(t)PnEnSn(s)Pnx

= lim EnSn(t

n—• oo

+ s)Pnx = S0(t + s)x

n—^oo

and use the estimate \So(t)So(s)x —

EnSn(t)Pn

< \S0(t)S0(s)x

-

EnSn(t)PnS0(s)x\

+ \EnSn{t)PnS0(s)x < \S0(t)S0(s)x

-

EnSn(t)PnEnSn(s)Pnx\ + Me w t |5 0 (s)x -

- EnSn(t)PnS0(s)x\

EnSn(s)Pnx\.

For all A with Re A > LJ, we have (see Proposition 2.20) I(XI - Ao)'^

- En(\In

- An)-lPnx\

< /

e-KeXt\S0(t)x

- EnSn{t)Pnx\

dt.

Jo

Using Re A > w and Lebesgue's dominated convergence theorem we conclude that lim„_>oo |(A7 — A0)~1x — En(\In — An)~1Pnx\ = 0 for all x G X and all A with Re A > u. D The last part of Lemma 4.1 suggests that convergence of Co-semigroups is connected with convergence of the resolvents of the corresponding infinitesimal generators. This is essentially the content of the Trotter-Kato approximation theorem. We first state a version which stresses this point and is very convenient for applications. 4.2. Theorem (Trotter—Kato). Suppose that assumptions (al) and (a3) are satisfied. Let S(-) and Sn(-), n = 1,2,..., be Co-semigroups on X resp. Xn of class G(M,u) with infinitesimal generators A resp. An. Then the following two statements are equivalent: (i) There exists a A0 G p(A) n H^Li p ( A 0 such that, for all x G X, \En(X0In - An)~lPnx

- ( A 0 7 - ^4)—1ar| ->• 0

as

n^-oo.

(ii) For every x G X and t > 0 we have lim \EnSn{t)Pnx

- S(t)x\ = 0

where the convergence is uniform for t in bounded intervals. If (i) or (ii) is true, then (i) holds for all A with Re A > u.

4.1. The Trotter-Kato

theorem

119

Proof, a) We first prove that (i) implies (ii). Without restriction of generality we can assume that Ao = 0 (otherwise we take A — A-Xol and An = An - XQI„ which generate the semigroups S(t) = e~XotS(t) and Sn(t) = e~XotSn(t)). For x € X we define the error en(t)x = Sn{t)Pnx

- PnS(t)x,

n = 1,2,..., t > 0.

We fix T > 0. The operators e„(i) are uniformly bounded for t G [0,T] and n = 1,2,... . For x e dom A the functions « n (-) defined by un(t) =A-len{t)x,

n= 1,2,..., t > 0,

1

are in C (0, oo;X„) and satisfy (t) + (4.1) v

PnAnAS(t)x,

, ,

;

« B (o) = o,

where A n — A - 1 - E n A ~ 1 P n . Indeed, J 4~ 1 S„(i)P n a; = S n ^ A ^ P n a ; is continuously differentiable on [0, oo), because A~ : P„x is in dom An, whereas PnS(t)x is differentiable, because x G dom A and A~ 1 P„ is bounded. Equation (4.1) follows by simple computations. By the variation of parameters formula we get from (4.1) the representation un(t) = / Sn{t-T)PnAnAS(r)xdT, t > 0, xedomA. Jo If we assume x G dom A2 then r —> A 5 ( T ) X is differentiable and integration by parts gives (using also Sn(t — r ) = A~ 1 A„5„(t - r ) = —A~1(d/dt)Sn(t - r)) un{t) = -A-lPnAnS(t)Ax

+

A^Sn^PnAnAx

+ A*1 I Sn(t - T)PnAnS(T)A2xdr, Jo

t > 0, xe dom A 2 .

This implies en(t)x = -PnAnS{t)Ax (4.2)

+

Sn{t)PnAnAx

Sn{t-T)PnAnS(T)A2xdT

+ / Jo

:= Dln + D2n + D3n, For any T > 0 the set {S(t)Ax strongly as n —> oo implies

t>0,

x£ dom A 2 .

| 0 < t < T} is compact. Therefore A„ -> 0

lim £>i„ = 0 uniformly on [0, T]. n—>oo

Since the operators Sn(t)Pn 1,2,..., we have also

are uniformly bounded for t G [0, T] and n =

lim Z?2« = 0 uniformly on [0,T\.

Chapter 4. Approximation

120

of

Co-Semigroups

For D3n we observe that {S{t)A2x | 0 < t < T} is compact and Sn(t — r)Pn are uniformly bounded for 0 < r < t < T and n = 1, 2 , . . . . Therefore An —> 0 strongly as n —> oo implies also in this case lim £>3„ = 0 uniformly on [0, T]. n—>oo

This finishes the proof that lim„_ >00 e„(i)a; = 0 uniformly on [0,T] for all x G dom A 2 . Since the operators en(t) are uniformly bounded for t G [0,T] and n = 1,2,..., we get by a standard density argument lim en(t)x = 0 uniformly on [0,T] n—voo

for all a; G X. It remains to prove lim \(EnPn - I)S(t)x\

= 0 uniformly on [0, T].

Since the operators Sn-P™ — I are uniformly bounded for n = 1,2,..., it is enough to prove that lim„_>.00(.E„P„ - I)x = 0 for all x G d o m A For an x G domA we have (observe that (EnPn — I)Enyn = 0 for yn G Xn) (EnPn - I)x = (EnPn - I)AnAx

->• 0 as n ->• CXD.

b) Assume now that (ii) holds. Then Lemma 4.1 immediately implies that (i) is true for all A with Re A > u>. • The approximation scheme given by the spaces Xn and the generators An is called stable if there exist constants M > 1 and to G K such that Sn(-) G G(M, w) for all n = 1,2,... . The approximation scheme is called consistent if condition (i) of Theorem 4.2 is satisfied and convergent if condition (ii) of that theorem holds. The Trotter-Kato theorem as stated above tells us that for a stable approximation scheme consistency of the scheme is equivalent with convergence of the scheme. However, Lemma 4.1 tells us that the Trotter-Kato theorem can be stated in a stronger form: For an approximation scheme convergence is equivalent to stability and consistency. In applications of Theorem 4.2 one encounters two major difficulties: a) In general it is very difficult to establish stability of the approximation scheme in cases, where M > 1 is required. The Hille-Yosida theorem (Theorem 2.22) tells us that ||5„(<)|| < Meut, t > 0, n = 1,2,..., if and only if A G p{An), n — 1,2,..., for Re A > w and _. M \\{An-\In) fe|| < —— - , R e A > o ; , n = l , 2 , . . . and fc = 1,2,... . (Re A — w)K To establish these estimates is in general very difficult if not impossible. A convenient way to verify stability of an approximation scheme is to use the

4.1. The Trotter-Kato

theorem

121

Lumer-Phillips theorem (Theorem 2.25) and to establish dissipativity estimates possibly after renorming the spaces Xn with uniformly equivalent norms. b) In order to prove consistency of an approximation scheme one in general tries to avoid computation of the operators (\In — An)~x in order to verify condition (i) of Theorem 4.2 directly. However it is rather easy to compute explicit representations of the approximating operators Ara. Therefore one would like to replace condition (i) by a condition involving convergence of the operators An to A in some sense. 4.3. Proposition. Let assumptions (al) and (a3) be satisfied and assume that the semigroups Sn(-) generated by the operators An are of class G(M,OJ) for some fixed constants M > 1, w 6 1 . Then condition (i) of Theorem 4.2 holds if and only if (a2) and the following two conditions are satisfied: (cl)

There exists a subset D c domA such that D — X and (X0I - A)D = X for a A0 > w.

(c2)

For all x € D there exists a sequence ~xn € dom A n , n = 1,2,..., such that lim EnXn = x

and

lim J5„A„:z?„ = Ax.

Proof. Without restriction of generality we can assume that Ao = 0. a) We first prove that condition (i) in Theorem 4.2 implies (a2), (cl) and (c2). We choose D = domA, which implies AD = X, i.e., (cl) is satisfied. In the proof of Theorem 4.2 we have already shown that (i) implies (a2). We next fix a; £ domA, choose y € X with x = A~ly and set ~xn = A^PnAx. Then we have En'Xn — x = EnA~ Pny — A~1y —• 0 as n —> oo by (i). Furthermore, we have using (a2) EnAnTn-

Ax = EnAnA^Pny

- AA~xy = EnPny ~y -» 0 as n -> oo,

which shows that (c2) is also true. b) In order to show that (a2), (cl) and (c2) imply condition (i) of Theorem 4.2 we use the identity (4.3)

EnA~lPn

- A'1 = En{A-xPnA

- Pn)A~l

+ (EnPn -

For x 6 AD we choose a y € D with x — Ay and set yn = A^-Pnx Let ~xn be a sequence for y according to (c2). Then we have \x„ - Pny\n = \Pn(Enxn

- y)\n < Mi\Enxn

I)A~\ =

A^xPnAy.

- y\ -» 0 as n -» oo

122

Chapter 4. Approximation

of

Co-Semigroups

and \%n

Vnln 5: \\An II \-™n%n ~

rnAy\n

1

<

\\A- \\\\Pn\\\EnAnxn-Ay\

which tends also to 0 as n -»• oo. Note that (||A~ 1 ||)„ = i :2 ,... is uniformly bounded. The last two estimates prove that \Pny-yn\n

<\PnV-xn\n

+ \xn-yn\n-+0

as n ->• oo.

This estimate together with (4.3) and (a2) implies \EnA-xPnx

- A - ^ l < \En(yn - P n B ) | + |B n F n ? / - y\ <M2\

+ \EnPny — y\ ->• 0 as n -> oo

for all x G AD. A simple density argument finishes the proof (note that ||S n i4~ 1 P„|| is uniformly bounded with respect t o n ) . • Lemma 4.1 suggests the conjecture that in the Trotter-Kato theorem one only needs to have strong convergence of the resolvent operators (XIn — An)^1 to a bounded linear operator in order to conclude the existence of a limiting semigroup. That this is really the case is the objective of the considerations below. The approach we follow goes back to J. Kisyhski (see [Ki]) and can be extended to nonlinear semigroups (see [Le-Rei]). A basic element in this approach is the following characterization of strong convergence of bounded linear operators as boundedness of a linear operator in an appropriately defined product space. We assume that (al) - (a3) are satisfied. Let X°° be the linear space of all sequences x = (xn)ne^ with xn G Xn, n = 1,2,..., such that XQ := linin-yoo Enxn exists with XQ G X.1 With the norm | x | x ~ =sup|a; n |„. X°° is a Banach space (note that X is a closed subspace of Z). For linear operators Cn : domC re —> Xn on Xn, n — l,2,...,we linear operator C : dom C —>• X°° on X°° by

define the

dom C = {x G X°° | xn G domC n ,ji = 1,2,... ,and (4.4) v

'

lim EnCnxn n->oo

exists in X\, '

r x e

Cu — (Cnxn)neH f° dom C. 4.4. Lemma. The following two statements are equivalent: (i) The operators Cn are bounded on Xn, n = 1,2,..., and limn-voo EnCnPnx exists in X for all x G X. 1 For elements x, y, ... in X°° we shall denote the corresponding sequences by (xn)n6Ni (?/n)neN> • • • a n d t n e limits limji-Kx, E„xn, limn^oo Enyn, . . . by x0,yo, ••• •

4.1. The Trotter-Kato theorem

123

(ii) The operator C defined by (4.4) is bounded on X°°. If (i) or, equivalently, (ii) is true, then CQX = \im.n^00EnCnPnx, defines a bounded linear operator on X.

x £ X,

Proof, a) Assume first that (i) is true and define Co as above in the lemma. Obviously Co is bounded on X. By the uniform boundedness principle we see that ||E n C n P n || < a, n = 1,2,... for some constant a > 0. For any x £ X°° we get the estimate \EnCnXn

— Co^ol < \EnCnPnEnXn

< a\Enxn

— EnCnPnXo\

- %o\ + \EnCnPnxo

+ \EnCnPnXo

- C0x0\ -» 0

— CQXQ\

as

n->• oo,

which proves x £ dom C, i.e., we have dom C — X°°. The estimate |C„a;„| = \PnEnC„PnEnxn\n < MiM2a\xn\n < M 1 M 2 a|cc|x~ for x £ X°° shows that C is bounded (with ||C|| < MiM2a). b) Let (ii) be true. For fixed t i e N and arbitrary xn G Xn we set Xk = 0 for A; G N \ n . Then we have x = {xk)ken G X°° = dom C. By definition of dom C we get xn G d o m d . Furthermore we have |C n a; n | n = \Cx\x°° < \\C\\ \xn\n, i.e., Cn is bounded on Xn, n = 1, 2 , . . . . For arbitrary x £ X we set a;„ = P„a;, n = 1,2,..., which by (a2) implies (x„)neN G X°° = dom C. By definition of dom C, lim„_>oo EnCnPnx exists in X. • 4.5. Proposition. Assume that (al) - (a3) hold. Let An be infinitesimal generators of Co-semigroups Sn{-) on Xn, n = 1,2,... . Then the following two statements are equivalent: (i) So(t)x := lim„_+00 EnSn(t)Pnx exists in X for all x £ X and t>0, the convergence being uniform on bounded t-intervals. (ii) The operator A defined by (4.4) corresponding to the operators An, n — 1,2,..., is the infinitesimal generator of a Co-semigroup on X°°. If (i) or equivalently (ii) holds, then So(-) is a Co-semigroup on X and its infinitesimal generator Ao is given by dom Ac = { i € X | there exists an x £ dom A with xo = x}, AQX = lim EnAnxn

for x G dom Ao, where x is any element

n—»oo

in dom A with XQ = x. Proof, a) Suppose first that (i) is true. By Lemma 4.1, 5o(-) is a Co-semigroup on X and we have Sn(-) £ G(M,UJ), n = 0 , 1 , . . . , for some constants M > 1, w e l . Let A denote the infinitesimal generator of SQ(-). Moreover, we have (4.5)

lim En(XIn - A^PnX

= (XI-

A)'lx,

x G X, Re A > UJ.

The operators S(t), t > 0, defined by (4.6)

S(t)x = (Sn(t)xn)neN,

t > 0, x £ X°°,

Chapter 4. Approximation of Co-Semigroups

124

are bounded on X°° according to Lemma 4.4. The semigroup property follows easily from (4.6). It remains to prove strong continuity of S(-) at zero. We set M = MXM2M (i.e., \\EnSn{t)Pn\\ < M e " ' , t > 0) and fix x € X°°. Next we choose N G N such that \Enxn - xQ\ < -^—(1 + Me")'1

and

oN±\

\EnSn{t)Pnx0

- S0{t)x0\ < -^— oNl\

for n > TV and 0 < t < 1. Then we choose S G (0,1] with \S0(t)x0 - x0\ < TTT

and

\S„(t)xn - xn\n < e

for 0 < t < 6 and n = 1 , . . . , N - 1. For n> N and 0 < t < S we get \Dn[t)Xn

%n\n S \*n£Jnd
-* n-^n^n\,*J-* n ^ 0 | n

+ \PnEnSn(t)PnX0 + \PnS0(t)x0

-

PnSo(t)x0\n

< Mj(l + Me w *)|£ n a: n - x 0 | + | + | < e. Thus we have shown that \S(t)x — x\ < e for 0 < t < S. Let B denote the infinitesimal generator of the semigroup S(-). We choose x G d o m B and set y = Bx. From y — lim/40 h~1(S(h)x — x) we conclude lim^o h~1(Sn(t)xn — xn) = yn for n = 1,2,... . This implies xn G domj4„ and Anxn = yn, n = 1,2,... . From y G X°° we get y0 = l i m ^ ^ Enyn = lim„^oo EnAnxn G X. This shows x G domj4 and y = Ax, i.e., A is an extension of B. Let x G dom A be given and choose A with Re A > u>. We set y — Ax = {Anxn)neN- For z„ = (A7n - An)xn, n — 1,2,..., we get l i m ^ o o E n z n = lim„_>00(AJ5„a;„ - EnAnxn) = \x0 - yo G X. This shows that z = (zn)n€^ G X°°. From Proposition 2.20 we get / e~xtS{t)z dt = (XI-B^ze dom B. Jo Using strong continuity of S(•) at 0 it is easy to see that (4.7)

(4.8)

/

e~xtS(t)zdt=

Jo

lim wN, JV->oo

where

wN = («,W) BeN = 1 £ AT

e -^

fe 1

- )/ JV 5((fc - 1)/AT),.

fe=i

We have AT

1 £ e-^-^NSn((k N

k=i

- l)/N)zn,

n = 1,2,...,

4.1. The Trotter-Kato

125

theorem

and /«oo

lim wnN) = / JV-s-oo

e-xtSn{t)zndt

= (XIn - A„)~ 1 z„ = xn,

n = 1,2,... .

70

This together with (4.7) and (4.8) implies x = (A/-J5)_1z GdomB and Bx = Xx — z == (Aa;„ — 2:n)raeN = {Anxn)ne^ = Ax by definition of zn. Thus we have shown that A — B, i.e., >1 is an infinitesimal generator. b) For arbitrary x G X we choose x G dom .A with XQ = x and set y = Az, z = (XI - B)x. Observing \\En(XIn - A„)- 1 P„|| < M1M2M(ReX - w)" 1 , n = 1,2,..., we get (using also (4.5)) \En(XIn - An^Zn

- (XI - Ao) _1 ^o|

< \En(XIn — An) + \En(XIn

PnEnzn

- An^PnZO

< ^ 2 M \ E n Z n Re A — w

— En(XIn — An) ~ (XI - Ao)'1

- Z 0 | + \En(XIn '

- A ^ '

Pnzo\

Zo\ 1

PnZ0

- (XI - Ao)^

On the other hand we have lim„_>oo En(XIn — An)~1zn so that xo = (XI -

AQ)~1ZQ

Z0\ - > 0. '

= rim„_).00 Enxn = xo,

G domAo

resp. XXO — AXQ = z0 = XxQ — yo. This implies Axo = yo = limn-joo EnAnxn for any x G dom A with XQ = x. This proves that the definition of the operator Ao in the proposition makes sense and that the infinitesimal generator A of So(-) is an extension of AQ . In order to prove A = Ao we choose an xo G dom A and set yo = (XI—A)xo, xn = (XIn - An)~1Pny0 G d o m ^ „ . By Lemma 4.1 we have l i m ^ o o Enxn = (XI - A)-xyo = x0, i.e., ( X „ ) „ € N G X°°. From E „ A n x n = EnAn(XIn

- An)"lPnVQ = En(X(XIn

- An)'1

= XEnxn - EnPnyo,

-

In)P„y0 n = l,2,...,

we obtain (using also (a2)) lim EnAnxn

= Xx0 - yo = Ax0.

n—too

This proves that x G dom^4, XQ G dom^o a n d ^0^0 = AXQ. Consequently we have A — AQ. c) Assume that (ii) holds and denote by S(-) the Co-semigroup on X°° generated by A. For some constants M > 1 and UJ € R we have ||S(i)|| < M e u t , t > 0. Moreover, (J - / i A ) _ 1 exists for 0 < ]i < l/|w|. For x G X°° we set y = (yn,k)neN = {I — fiA)~kx, k = 1,2,... . It is easy to see that

126

Chapter 4. Approximation

Vn,k = (In — nAn)~kxn, rem 2.23) we have (4.9)

Co-Semigroups

n = 1,2,... . By the exponential formula (see Theo-

S(t)x = lim (i - 7 A ) " x,

We set S(t)x = (yn(t))neN yn,k = (In - (t/k)An)~kxn.

of

x G X°°.

and (/ - (t/k)A) x = (yn,k)nefi- Then we have This and (4.9) imply for k -> oo

yn(t) = Sn(t)xn,

t > 0, n= 1,2,... .

We define y0(t) = lim^oo Enyn(t) = lim^oo EnSn(t)xn and fix a T > 0. Continuity of S(-)x implies equicontinuity of the sequence (£'Tlo7l(*)icn)ri^^. Since (EnSn(t)xn)nen is a convergent sequence for any t e [0, T], it is precompact. By Ascoli's theorem (see for instance [Di, p. 137]) we get a subsequence which is uniformly convergent on [0,T]. Since the limit function necessarily is yo(-), we see that yo(-) is continuous on [0,T]. Given e > 0 we choose 6 > 0 such that fA-,^ (4.10)

\S(h)x-Sfo)x\x°°
and

\y0(ti) - y0(t2)\

o for t 1 , i 2 £ [0,T] with \ti

< | 3

-t2\<S.

For any r e [0, T] there exists an NT G N with |j/o(r) - £ , n 5 n (r)a; n | < - for n > NT. We set U = iS for i = 1 , . . . ,k — 1 and £*, = T, where fc is determined by (k — 1)6 N := maxj=ii...ifc 7Vti the estimate \EnS„(t)xn

- y0(t)\ < |

(t)xn -

EnSn{tj)

+ \EnSn(tj)xn ^ e ^3

e +

- y0{tj)\ + \y0(tj) -

y0(t)\

e

3+ 3

= £

'

where we have also used (4.10). This proves lim„_>.00 EnSn(t)xn = y0(t) uniformly on [0,T]. From \\EnSn(t)Pn\\ < MxM2Mew\ t > 0, n = 1,2,..., we obtain \EnSn(t)Pnx0

- EnSn{t)xn\

< M1M2Meu,t\Enxn

- x0\,

from which we conclude that also lmin^oo EnSn(t)PnXQ

[0,71.

n = 1,2,...,

= yo(t) uniformly on

•

We are now in a position to prove the version of the Trotter-Kato theorem which assures also existence of a limiting semigroup.

4.1.

The Trotter-Kato

theorem

127

4.6. Theorem. Assume that (al) - (a3) hold and let An, n = 1 , 2 , . . . , be the infinitesimal generators of the Co-semigroups Sn(-) on Xn. The following two statements are equivalent: (i) So(t)x := limn^oo EnSn{t)Pnx exists in X for all x e X and t > 0, the convergence being uniform for t in bounded intervals. (ii) a) There exist constants M > 1 andw G R such that Sn(-) G G(M,w). b) R(X)x := lim„_>oo En(XIn — An)~1Pnx exists in X for allx G X and all X with Re A > w. c) range R(X) is dense in X for all X with Re A > w. If (i) or, equivalently, (ii) is true, then R(X) = (XI — AQ)-1, ReA > AQ is the infinitesimal generator of the Co-semigroup So(-) on X.

LO,

where

Proof, a) First assume that (i) holds. Then Lemma 4.1 implies that So(-) is a Co-semigroup on X with infinitesimal generator AQ. Moreover, (ii) is true with R(X)x = (XI - Ao)~lx. b) Let (ii) be satisfied. We want to show that the operator A as defined in (4.4) is the infinitesimal generator of a Co-semigroup on X°°. For arbitrary x G X°° and A with Re A > LJ we consider the equation

(XI — A)y = x, This is equivalent to (A/„ — An)yn

yedomA.

— xn, n— 1,2,..., and by (ii), a) to

Vn = (XIn - An)~1xn,

n = l,2,....

1

Using || J E„(A/„-A r l )- P„|| < M 1 M 2 M ( R e A - w ) - 1 , n = 1,2,..., and (ii), b), we have lim Enyn = lim En(XIn - A„) _1 a;„ = lim En(XIn - An)~lPnxo n—>oo

n—foo

= R(X)x0,

n—>-oo

i.e., yo = R(X)xo and consequently y G X°°. The equation lim EnAnyn

= lim En(X(XIn

- A„) _1 a;„ - xn)

= lim (XEnyn - Enxn)

= Xy0 - x0

n—foo

proves that y G dom.4. Thus we have shown that {A | ReA > w} C p(A) and (A7 — A)~^x = ((XIn — An)~1Xn)neN for x G X°°. It is easy to see that

"^-^"^(dSb)'

ReA>w

-

This implies also that A is closed. By induction we see that H ( A J - A ) - f c | | < ( R e A M ^ , fc = l , 2 , . . . , R e A > W .

Chapter 4. Approximation

128

of

Co-Semigroups

In view of the Hille-Yosida theorem it remains to prove that dom A is dense in X°°. For x G X°° and e > 0 we choose an Ni G N such that \Enxn -x0\

< ——

forn>iVi.

By condition (ii), c), we can choose yo € range-R(A), y0 = R(\)a, e \Vo ~ x0\ < 3Mi We set yn = (A7n — An)~lPna lim Enyn n—KXJ

with

and see from condition (ii), b), that

lim En(XIn - A~lPna

= R(X)a - y0,

n—• oo

which shows (j/n)neN G X°°. From a = l i m ^ o o EnPna = lim„^. 00 £„(A7„ An)yn and lim^^co i?n?/„ = yo w e conclude that linin^oo EnAnyn exists (and equals Aj/o — o). Consequently we have (yn)neN e dom .A. We choose ^ e N such that l-Bnj/n - j/ol < TTT-

for n > AT2.

The estimates given above show that, for n > N := max(iVi, .W2), we have the estimate \Xn

yn\n ~ \i-n\-^n^n

< Mx(\Enxn

^nVn^n

- xQ\ + |x 0 - y0\ + |j/o - Enyn\)

< e.

For n = 1 , . . . , N — 1 we choose y„ € dom An such that |?/„ — xn\n < e and set y-a = 2/n for n> N. Then we have y = {yn)nefi € dom .A and |sc — y\x<*> < e. According to the Hille-Yosida theorem A is the infinitesimal generator of a Co-semigroup on X°°. Then Proposition 4.5 implies (i). • The following lemma shows that condition (ii) in Theorem 4.6 needs only to hold for some A with Re A > UJ (compare also condition (i) in Theorem 4.2). 4.7. Lemma. Assume that (al) - (a3) hold and let An, n = 1,2,..., be the infinitesimal generators of the CQ - semigroups Sn(-) on Xn. Moreover, assume that there exist constants M > 1 and UJ G R with 5„(-) G G(M,UJ), n = 1,2,... . Then the following two statements are equivalent: (a) limn^oo En(XIn — An)~1Pnx exists in X for all x G X and all A with Re A > UJ. (b) There exists a Ao with Re Ao > UJ such that limn^oo En(XIn — An)~1Pnx exists in X for all x G X. If (a) or, equivalently, (b) is satisfied, we set R(X)x = limn^oo En(XIn — An)~1Pnx, x G X, Re A > w. Then range R{X) = X for all X with Re A > w if and only if range R(XQ) = X for some AQ with ReAo > UJ.

4.1. The Trotter-Kato theorem

129

Proof, a) Assume that (b) is satisfied. We set A = { A e C | R e A > w and lim EJXIn

-

AnylPnx

n~>oo

exists in X for all x £ X } . We fix A € A. Then using ||(AJ„ - A n )- f c || < M(ReA - w)~fc, k = 1,2,..., we have (/*/„ - A^-1

= (XIn - An)~l

(J„ + (,x - A)(A7„ - A , , ) - 1 ) *

OO

(4.11)

k=0

for /i with | A — n\ < Re A — w. We fix e > 0 and x € X. Then there exists an N € N such that

(A - /x)fc(A/„ - A J - * - 1

I £

<

fc=AT+l

for all n — 1,2,

AMiM2\x

This implies En\ixln~

An)

PnX — him\ll,lm

— Am)

PmX

TV

< E iA - ^ife ^ ( A / « - ^ r f e - 1 P n z

(4.12)

fe=0 +

2 l

for all n, m G N. Since we have A e A, it follows that \En{\In — An)~ Pnz Em(XIm ~ Am)~1Pmz\ —> 0 for all z e X as m, n -4 oo. Assume that y = limn^oo En(XIn - A„) _ f c P„x exists in X. Then we get \J^n{Xln — An) < \\Bn(XIn

*n%

- A „ ) - 1 P „ | | \En(XIn

Am)

~ An)-lPn\\

\Em{XIm

+ \En(XIn

- An^Pny

- Em(XIm

MXM2M,

1

PmX\

- An)-kPnx

+ \\En(XIn

+ \\Em(XIm

<

&m\Xlm

- Em(XIm

-

- A m ) ~ * P m X - y\ - Am)'1

- A m ) - P m | | \y - Em(XIm

-

Pmy\ Am)~kPmx\

En(XIn — An) Pnx — Em\Xlm — Am) Pmx\ ReA-w 2MiM 2 M V — Em{XIm — Am) Pmx\ Re A — u + \En{XIn ~ A n ) - 1 F „ y - Em{XIm - A ™ ) " 1 / ^ ! -»• 0

Am)-kPmx\

Chapter 4. Approximation

130

for m,n —> oo. By induction we see that lim„_>oo En(\In in X for all fceN. We choose N0 G N such that \En(XIn — An)

Pnx — Em{Xlm

of

Co-Semigroups k

— An)

Pnx exists

Pmx\

— Am)

e 2(A + 1) max fc=0 ,..., w (Re A - u)k r

for k = 0 , . . . , N and n,m, > No. Using this in (4.12) we get, for |A — /i| < Re A — u>, \En(fj,In - An)~ Pnx - Em(^Im

- Am)" Pmx\ < - + - — e,

n,m>

N0.

This implies that lim^^co En(fj,In — A „ ) _ 1 Pnx exists in X for \X~n\ < Re A - w and x e X, i.e., // G A for |A - fi\ < Re A - w. This proves that A is open. Let \i € {A G C | Re A > w} n A, i.e., /i is a cluster point of A relative to the open right-half plane Re A > w. Consequently there exists a A G A with |A-/x| < ( R e / i - u ; ) / 2 . This implies Re A — o> = Re /i - w - Re(// - A) > |/x — A| and consequently / i e A i n view of the considerations given above. Thus A is also relatively closed in Re A > u, which finishes the proof of A = {A G C | Re A > <JJ}.

b) Assume that (a) or, equivalently, (b) holds. In order to prove the last part of the lemma it suffices to prove that for any A with Re A > UJ and range 72(A) = X there exists a neighborhood U with range R(n) = X for all fi G U. Assume that range R(X) = X for a A with Re A > w. For |/J - A| < Re A - u; we have oo

En(ln

+ (IM - A)(A/„- J 4 T l )- 1 )- 1 P„ = ]T(A - n)k(En{XIn

An)-lPnf

-

k=0

-I

+ EnPn

= (I + (H- X)En(XIn - An)-xPn)~l

-1 + EnPn

and consequently (note that PnEn — In) En(nln

- An)-lPn

= En(ln

+ (/x - A)(A7„ - A„)- 1 )" 1 (A7„ -

An)~1Pn

= (l + (fi- X)En(XIn - J 4„)- 1 P„)" 1 E„(A/„ + (7 - EnPn)En{XIn

An)-lPn

- A„)" 1 P„

= (I + (/i - A)£„(A7„ - A„)- 1 P„)" 1 E n (A7„ -

An)~lPn.

4.2. Approximation of nonhomogeneous problems

131

This and the estimate || (J + (M - A)£„(A/„ - A „ ) - 1 P „ ) " 1 |

/ + J2(X - ii)kEn{\In - An)~kPn k=\

fc=l

for | A — /i| < Re A — ui, n = 1,2,..., imply fl(A) (/ + (/i - A)JJ(A)) _1 a; - £„(/*/„ - A x J - ^ n i < cA,„| ( ( / + (/i - A)E n (A/ n - A n )- x P n )i?(A) - E n (A/„ - A n ) - 1 P „ ( 7 + (^ - A)fl(A))) (/ + (/i - A)i?(A)) _1 x for x 6 X and \fi — A| < Re A — w. The right-hand side tends to 0 as n —> oo. Thus we have shown that lim £„(/x/n - An)'lPnx

= i?(A)(J + (/i - A)P(A)) _1 a;.

On the other hand we have limjj-yoo En{^iln — An)~lPnx 1

R(fi) = R{X)(I + (fj,- X)R{\))" ,

= R(fi)x. This proves

\n - A| < ReA - w.

It is clear that range(7 + (/* — \)R(X))~1 = X for |/x — A| < Re A — w. Then we get immediately from range R(X) = X that also range i?(/i) = X for |/x — A| < ReA-w. •

4.2.

Approximation of nonhomogeneous problems

In this section we consider approximation of the nonhomogeneous problem u(t) = Au(t) + f(t),

(4.13)

t > 0,

u(0) — x, 1

where x 6 X and / € i ( 0 , oo; X) are given and A is the infinitesimal generator of a Co-semigroup S(-). As in Section 4.1 we assume that X is a closed subspace of a Banach space Z and that a sequence Xn, n = 1,2,..., of Banach spaces together with bounded linear operators Pn : Z —> Xn and E„ : Xn —• Z are given. We assume that (al) - (a3) hold. Together with (4.13) we consider the approximating problems Un(t) = AnU„(t) + /„(*), 4.14 un(0) = xn,

t> 0,

132

Chapter 4. Approximation

of

Co-Semigroups

where xn G Xn, fn £ L 1 (0, oo;X ra ) are given and the An are infinitesimal generators of C o-semigroups *S'n(*) on Xn, n — 1,2,... . The mild solutions to problems (4.13) and (4.14) are given by u(t) = S(t)x + un{t) = Sn(t)xn

Jo

S(t-

T)f(r) dr,

+ f Sn(tJo

t > 0,

T)fn(r) dr,

resp. by

t > 0.

If we assume that condition (i) of Theorem 4.2 holds and \Enfn — /|LI(O,OO;JV) ~• 0 as n ->• oo, then it is easy to show that, for xn — Pnx, we have Enun{t) -¥ u(t) as n —^ oo for any t > 0. One only has to use Lebesgue's dominated convergence theorem. However, we want to prove that this convergence is uniform for t in bounded intervals. In the following we show that this can be accomplished by converting the nonhomogeneous Cauchy problem (4.13) into a homogeneous problem in some product space. Set Z = Z x L^O.oojZ) and X = X x V-(Q,oo;X) with the norm \(x,f)\z = \x\ + | / | i i , (x,f) G Z. We define the operators <S(-) by S(t)(x,f)=(s(t)x

+ Js(t-T)f(T)dT,ft),

i>0,

(x,f)eX,

where we define / ' by / ' ( r ) = f(t + r ) , r > 0. 4.8. L e m m a , a) The operators S(t), t>0,

constitute a Co-semigroup on X.

b) If S(-) is in G{M,w), then we have S(-) £ G(M, max(w,0)). c) Let A denote the infinitesimal generator of S(-). Then A is given by dom A = dom A x W1'1 (0, oo; X), A{x, f) = (Ax + /(0), / ' ) ,

(x, f) e dom A.

Proof. It is very easy to see that <S(-) is a Co-semigroup on X, which satisfies statement b). Therefore we shall only give the proof for statement c). We choose (x,f) G dom .4 and fix A > max(w,0). Then we have, for some {V, 9) G X, /•OO

(x,f) 4

( -!5)

= (XI-A)-1(y,g)= , ,oc , j™e~xt(S(t)y

/

xt

e-

S(t)(y,g)dt

Jo

+ JQS(t-

r)g(r) dr) dt, J^e^g'dt)

.

4.2. Approximation

of nonhomogeneous

problems

133

This implies /•oo

(A

/*00

e-Mgt{T)dt=

f(r)= JO

1 R \

(4-16)

e-xtg(t

+ T)dt

JO

oo = e A r / e~xtg{t) dt,

r > 0,

which proves / G VF1'1(0, oo;X). From (4.15) we get also x=

/ e- A t 5(t)?/di+ / e~At / Jo Jo Jo /•oo

={\I-A)-1y+

(4.17)

S(t~T)g(T)dTdt

/»oo

e~XtS{t) e~Xr g{r) dr dt Jo Jo

= (\I-A)-l(y+

J e~XTg{r) d r ) £ dom A.

This together with (4.15) and (4.16) proves {x,f)£domAxW1'1{0,oo;X)

and ^ ( i , / ) = {Ax + / ( 0 ) , / ' ) -

This proves that the operator B defined by dom 2? = dom A x VF1'1(0, oo;X) and B{x, f) = {Ax + /(0), / ' ) for {x, / ) G dom 6 is an extension of A. In order to prove B — A we fix A > max(w, 0) and assume that, for some {x, / ) G dom B, we have {XI — B){x,f) = 0. This is equivalent to (4.18a)

{XI-A)x-f{0)=0,

(4.18b)

Xf - f = 0.

Equation (4.18b) is equivalent to / ( r ) = e A r /(0), r > 0. In view of A > 0 we can have / G L 1 (0,oo;X) if and only if /(0) = 0. Consequently (4.18b) is equivalent to / = 0. Then (4.18a) is equivalent to {XI — A)x = 0 respectively to x = 0, because we have X > u>. Thus we have shown that XI — B is injective. Let (x, f) £ dom 23 be given. Since we have range(A/ — A) = X, there exists an {x, f) G dom.4 with {XI - B){x, / ) = (A7 - A){x, / ) = {XI - B){£, / ) . By injectivity of A J - B we get {x, f) = {x, / ) G dom A and B{x, f) ~ A{x, / ) . • From (4.15) - (4.17) we get the following representation of the resolvent operator (A — A)-1, Re A > max(w, 0): (4.19)

{XI-Ar'^g) = {{XI - A)'1 (y + y ° ° e - A ^ ( T ) dr), eA' J°°e-X'g{a)

for

da)

(y,g)eX. We can now state and prove the approximation result for nonhomogeneous Cauchy problems:

134

Chapter 4. Approximation

of

Co-Semigroups

4.9. Theorem. Assume that conditions (al) - (a3) hold and that A resp. An, n = 1,2,..., are infinitesimal generators of Co-semigroups S(-) on X resp. Sn(-) onXn. Furthermore, assume that the following assumptions are satisfied: (a) There exist constants M > 1 and ui G R such that S(-) G G(M,w) and Sn(-) £ G(M,to), n = 1,2,... . (b) The functions fn G L1 (0, oo; Xn), n = 1,2,..., satisfy lim \Enfn - f\Li = 0. (c) For any x 6 X and some A iw£/i Re A > max(u;, 0) we have lim \En(\In

- (XI - A)~lx\

- A^PnX

= 0.

TTien we have, for any x € X, lim Enun(t)

= u(t),

t>0,

wftere ifte convergence is uniform for t in bounded intervals andu(-) resp. un(-) denotes the mild solution of (4.13) resp. of (4.14) with xn = Pnx. Proof. The inequality f EnSn{t Jo

- T ) / „ ( T ) dr - f EnSn{t Jo

- T)Pnf{r)

dr

<

[\EnSn(t-T)Pn(Enfn(r)-f(T))\dr Jo < MiM^Me^lEnU - f\Li

together with condition (b) of the theorem shows that we can restrict ourselves to the case / „ — Pnf. Let the semigroups <£«(•) and their infinitesimal generators An be defined analogously as <S(-) and A. Then we have Sn(-) G G(M, max(w,0)), n = 1,2,..., from Lemma 4.8. We define the operators Vn : Z —>• Xn and £n : Xn^Zby Vn(x,f)

= (PnX,Pnf),

^ny^Wi Jn) ™ x^n^ni

•&njn))

(x,f)eZ \%ni Jn) t ™ni

where (Pnf)(T) = P „ / „ ( r ) and ( £ „ / „ ) (T) = Enfn{r), r > 0. Obviously we have ||P„|| < Mu \\Sn\\ < M2 and Vn£n = / . From \EnPnf - f\Li = J0\EnPnf{T) — f(r)\dT and Lebesgue's dominated convergence theorem we get also limn^oo £„Vn{x, f) = (x,f). Thus conditions (al) - (a3) hold for Vn and Sn. In order to verify condition (i) of Theorem 4.2 we use (4.19) (for A and An) and the estimate

|(A7 - A)-\x, f) - 8n(XIn - A O " 1 ? ^ , / ) !

4.3. Variational formulations

of the Trotter-Kato

(\I-A)~l(x

- A^-1

e - A T P „ / ( r ) dr)

(pnx + f

oo

poo

/

e-Xsf(s)ds-EnexA^x - En(XIn -

<\(\I-

135

f°°e-XTf{T)dr)

+ - En(\In

theorem

e-XsPnf(s)ds

J An)-lPnx\

poo

+ / e~XT\(XI - A)-lf(r) Jo /»oo

/»oo

eXr

+ /

An)-lPnf{T)\dr

- En(XIn -

e-Xs\f(s)

- EnPnf(s)\dsdr

=: I + II + III.

By assumption (c) we have limn-too I = 0. Again by assumption (c), the estimate |(A/ - A)~'f{r)

- En(\In

- An)-'Pnf(r)\

<

ffi±-^l^|/(T)|, IXB A — UJ

n = 1,2,..., r G [0, T), and by Lebesgue's dominated convergence theorem we prove that also limn-too II = 0. For III we get /•OO

m<

J

pOO

J

e-x{s-r)\f{s)-EnPnf{s)\dsdT

poo ps

/ / e- A < s " T W \f{s) - EnPnf(s) Jo Jo

| ds

< jJ°°\f(3)-EnPnf(8)\da. From lim„^oc EnPnf(s) = f(s) for s > 0 and \f(s) - EnPnf{s)\ < (1 + M\M2)\f(s)\ w e get that also limn->oo III = 0. Thus we have shown that condition (i) of Theorem 4.2 is satisfied. By this theorem we get that, for any

(x,f)ex, lira £nS„(t)Vn(x,

/ ) = S{t){x,f),

t > 0,

n—>oo

where the convergence is uniform for t in bounded intervals. Looking at the X-components we get the desired result. •

4.3.

Variational formulations of the Trotter—Kato theorem

In many applications the infinitesimal generator A of a Co-semigroup is defined via a sesquilinear form a defined on a dense subspace of X. In such a case it is natural to define the approximating operators by sesquilinear forms

Chapter 4. Approximation

136

of

Co-Semigroups

an approximating er. Therefore it is of interest to formulate stability and consistency of such an approximation scheme in terms of the sesquilinear forms a and an. Let A be the infinitesimal generator of the Co-semigroup S(-) on X and assume that (4.20)

Xn

is reflexive, n—1,2...

.

Whenever the approximation scheme given by the spaces Xn and the operators An represent a numerical approximation scheme for an abstract Cauchy problem, the spaces Xn will be finite dimensional and assumption (4.20) is trivially satisfied. Define a continuous sesquilinear form a on dom Ax Z* by (4.21)

a{x,x*) = (Ax,x*),

x G domA,

x* G Z*,

and let <7„, n = 1,2,..., be a sequence of continuous sesquilinear forms on Xn x Xn. The duality set of xn G Xn will be denoted by Fnxn. For each n let Dn be a Banach space isomorphic to Xn and in be a bounded isomorphism Dn ~* Xn. For the sequence (Xn,Dn,o-n,in)neN we formulate the following stability assumption: (Sv)

There exists a constant u> G K such that for every n G N and yn G Dn there exists an xn G Fn(inyn) with Rean(yn,xn)

< uj\i

The existence of approximating operators is assured by the following lemma: 4.10. Lemma. Assume that (4.20) and condition (Sv) are satisfied. Then, for each n G N, there exists a continuous linear operator An on Xn such that (4.22)

(Aninyn,

x*n) = an{yn, x*n) for all yn G Dn and xn G X*.

Moreover, An — wln is strictly dissipative. Proof. Since Xn is reflexive, there exists a continuous linear operator An : Dn -> Xn such that {Anyn, x*n) = o-n{yn, xn)

for all yn G Dn and x*n G X*.

Since in is bijective, it is continuously invertible. We define the operator An on Xn by An — Ani~l. Then it is clear that An satisfies (4.22). Condition (Sv) implies that for every xn = inyn G Xn there exists an x* G Fnxn with Re(Ana;„,a;*) < w|a;n|^, i.e., An — wln is dissipative. Since An is continuous, it is the infinitesimal generator of the Co-semigroup Sn(t) = exp(Ant) and therefore also strictly dissipative (see Theorem 2.25). •

4.3. Variational formulations of the Trotter-Kato theorem

137

For A > OJ and x G X we set y = (XI — A)~1x. From (4.21) we get (4.23)

\(y, x*) - a(y, x*) = {x, x*)

for all i ' e Z * .

Dissipativity of An — XIn implies (see Proposition 1.9) that (XIn — An)~l exists for A > OJ and \\(XIn - yl„) _ 1 || < (A — OJ)-1. The element yn = i~l(XIn — An)~lPnx satisfies HinVn,x*n) - a„{yn,xn)

= (Pnx,x*n)

for all x*n G X*.

This together with (4.23) implies that for arbitrary yn G D„, a;* G X£ and x* GZ* (4.24)

X(in(yn - yn),x*n) - an(yn - yn,x*n) = {Pnx,xn) -X((inyn,x*n)

-

(x,x*)

- {y,x*)) + an(yn,x*n) -

a(y,x*).

We introduce the following consistency hypotheses: (Cvl) There exists a dense subset D C domA such that (X0I - A)D — X for some Ao > w. (Cv2) For any y £ D there exist yn € Dn with lim \Eninyn

-y\=0

n—too

and, for any xn £ Xn, elements xn G Fnxn, lim

1

x*n G Z* with

(Pnx,x*n)-{x,x*n) - A o ( ( M / n , < > - (y,X*n))

+0-n(yn,Xn)-a(y,X*n)

= 0,

where x = (XQI — A)y. 4.11. Theorem. Assume that A is infinitesimal generator of a Co-semigroup S(-) onX and that Xn,Dn,in,an,En,Pn,n— 1,2,..., satisfy (al), (a3), (Sv) and (Cvl), (Cv2). Then, for every x e X, lim \EnSn(t)Pnx

- S(t)x\ = 0

n—too

uniformly for t in bounded intervals, where An is given by Lemma 4.10 and S„(-) denotes the Co-semigroup on Xn generated by the operator An. Proof. By Lemma 4.10 the operators An — wln are dissipative on Xn, which implies that Sn(-) is of class G(l,u>), n = 1,2,... . We choose y G D and set

138

Chapter 4. Approximation

x = (X0I - A)y and yn = C ^ W n - An)-lPnx. x = x^ we obtain ((X0In - An)in(yn

of

Co-Semigroups

From (4.22) and (4.24) with

- yn),x*n) = (Pnx,x*n) - {x,x*n) - X0{{inyn,x*n) + crn(yn,x*n)

-<j(y,x*n)

for all A0 > u>, yn G Dn, x*n G X* and x*n G Z*. Strict dissipativity implies (Ao - w)|i„(j/„ - j/„)£ < Re((A 0 i„ - An)in(yn <

- (y,x*n))

-

oiAn-u>In

yn),x*n)

\({XoIn-An)in(yn-yn),x*n)\

for all Ao > a; and a;J^ G Fn(in(yn - yn)). We choose yn G D n according to (Cv2). Then we get (setting xn = in(yn - yn) in (Cv2)) (4.25)

lim \in(yn - yn)\n = 0

Using inyn = (XoI„ - An)~lPnx \En{X0In - AnyYPnx

and

lim \Eninyn

- y\ = 0.

and (4.25) we have

- {X0I - A)~1x\ = \Eninyn

- y\

< \Enin(yn

- yn)\ + \Eninyn

- y\

< M2\in(yn

- yn)\n + \Eninyn - !/| -> 0 as n ->• oo

for all x G (Ao/ - A)D. Since (Ao/ — -A)£> is dense in X and \\En(XoIn A„) _ 1 P„ - (A 0 / - A)" 1 1| is uniformly bounded (by (MiM 2 + 1)(A0 - w)" 1 ), we get |-En(Ao/ra - An)~1Pnx

- (Ao/ - A) -1 ar| —• 0 as n ->• oo

for all x G X. The result follows now from Theorem 4.2.

•

4.12. Example. We consider the first order hyperbolic problem (4.26)

^u(t,x)

+ ^u(t,x)=0,

u(t,0) = 0,

0<x0,

t>0.

We study this problem in X = Z — Co(0,1), where Co(0,1) denotes the space of continuous functions on [0,1] vanishing at x = 0. The norm on X is the sup-norm. We first prove that the operator A defined by domA ={<j)£X\ =-(/>',

and '(0) = 0},

(f>£dom.A,

is the infinitesimal generator of a contraction semigroup S(-) on X. Set Y = {<j> G X | (j> = 0 on [0,6] for some 6 G (0,1]}, which obviously is a dense subspace of X. We choose eY and 5 G (0,1] such that <> / = 0 on [0,6].

4.3. Variational formulations

of the Trotter-Kato

theorem

139

For a sequence of mollifiers pn2, n = 1,2,..., we set n{x) = f_unPn{% ~ T)(J)(T)(1T, 0 < x < 1. It is easy to see that (/>„(0) = 0 for n > 1/8, i.e., <j>n G domA for n > 1/8, and that l i m ^ o o \<j> - ^„|oo = 0. Therefore domA is dense in Y and consequently also in X. In Example 2.24 we have shown that A is m-dissipative. By the Lumer-Phillips theorem we get the result. The numerical method analyzed here is the following first order finite difference scheme: (4 27)

Jtuk{t) = -£c{uk-i{i)-uk(t)),

fc

= l,...,n, t > 0,

«o(t) = 0, where uk(t) represents an approximation for u(t,xk) with a^ = fcAa:, Aa; = 1/n. We choose Xn = R n , n = 1, 2 , . . . , with norm |«| n = maxfc=i n |ufe|, u = ( u i , . . . , Wfc)T. Then the dual space is X* = Rn with norm |v|n>* = Ylt=i l^fcli i> G Mn. It is easy to see that, for u G X, we have w = ( i ^ , . . . , u„) G F n («) if and only if (4.28)

vk = ( H n S i g n Wi 10

f

°rfc = i' forfc^ i,

where i is an index with \uA = luL. We define n(x-Xk-i) Bk(x) = { n(xk+i - x) 0

for x G [xk-i,xk], for x G [xk, xk+1], elsewhere

and, for u = ( u i , . . . , un)T e X n resp. <j> € X, n

Enu = 'Y^ukBk,

Pn= ((xi),...,(xn)) .

*=i

It is easy to see that PnEn = In and ||P„|| = ||i? n || = 1, i.e., assumptions (al) and (a3) are satisfied. In order to show that (a2) also holds we observe that EnPn(j) is the first order spline interpolating <j> at the mesh points xk, k = 0,...,n (see [Schu, Theorem 2.6]; note that C 2 (0,1) n C 0 (0,1) is dense in Co(0,1) and \\EnPn\\ = 1, n = 1,2,...). We define the sesquilinear forms crn : Xn x X* —• R by 1 ™ CT„(U, V) = — J ^ ( « f e - l - Wfc)Vfc, x fc=i 2

p n is in <7g°(—1/n, 1/n) and satisfies pn > 0, f_\)

W G X „ , D G X*,

Pn{T)dr = 1.

140

Chapter 4. Approximation

of

Co-Semigroups

where uo = 0. For this example we can set Dn = Xn and in = /„. For u G Xn and v G Fn(u) (see (4.28)) we get 1 <Jn{u,v)

™

1

= -T—^iUk-l

-Uk)vk

= —

(m-!

-Ui)\u\n

s i g n Ui

fc=l

= -r-\u\n{ui-i

signui - \ui\) < 0,

i.e., the stability assumption (Sv) holds (with w = 0). In order to establish the consistency hypotheses we choose D = dom A, so that (Cvl) is automatically satisfied. For u G domA we define un G Xn by un = (u(xi),..., u(xn))J. Then Enun is the interpolating first order spline for u and we have \Enun — w|oo —» 0 as n —• oo. For ij)n G X n we set V>n = IV'nlnCi, where ej is the i-th vector of the canonical basis for Xn and i is chosen such that |("0n)i| = IV'nU- By the mean value theorem there exists a ( 6 {xi-\,xi) with
- u(xi))\ipn\n

=

-\ipn\nu'(£).

With such a £ we define Vn = IV'nln^ € X*, where 6% is the delta distribution with support at £. This implies (4.29)

* ( « , & ) = -lV»n|n«'(0 -

^n(Sn,C).

Simple computations show that (4.30)

\(un,rn)

and, for (f> = (4.31)

(\QI

\(Pn<j>,rn)

- ( « , C ) | = \u(xi)-u({)\

\tpn\n <

pUtl(l/n)\ipn\n

— A)u = \0u + u', - <
where pU]i resp. p ^ i denote the modulus of continuity for u resp. (f> on the interval [0,1] (see Appendix A.l). From (4.29) - (4.31) we see that (Cv2) holds. Therefore Theorem 4.11 implies lim 71—>0O

i{t,-)~Y,uk(t)Bk

0

fc=i

uniformly for t in bounded intervals, where u(t) = («i(£),... ,un(t)) solution of (4.26).

is the

4.4. An approximation

4.4.

result for analytic

semigroups

141

A n approximation result for analytic semigroups

In this section we show how sharp rate estimates for the approximations can be obtained in case of an analytic semigroup with a