A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point

A BEHAVIOR OF GENERALIZED SOLUTIONS OF THE DIRICHLET PROBLEM FOR

QUASILIENARELLIPTICDIVERGENCE

EQUATIONS OF SECOND ORDER NEAR A

CONICAL POINT M. V. Borsuk

UDC 517.956.25

The behavior of solutions of boundary-value problems for linear elliptic equations near corner, conical, and other irregular boundary points, is well known. The modern status of the theory of linear boundary-value problems in nonsmooth domains has been presented in the survey [i]. This can be appended by a recently published paper [2]. Also, during the last ten years, there has been developed the theory of regularity of solutions of elliptic boundary-value problems in nonsmooth domains for equations of second order [3-12]~ In [3-5] there has been discussed a generalization of solutions of the elliptic equation diva(Vu) = f(x), where a(p) is a strongly monotone coercion field of class C I. In papers [6-9] there was studied an equation for the p-harmonic (p > i) Laplace operator. There was considered positive generalizations of the solution of the Dirichlet problem and obtained almost exact estimates of the decrease rate, and also estimates of the gradient of the modulus of the solution in a neighborhood of a conical boundary point. As Tolksdorf [9, po 310, (iv)] has indicated, it would be desirable to establish such estimates regardless of the sign of a solution. In [i0] there was proved the solubility of the Dirichlet problem for the equation div(~.(Vu) V ~ ) ~ / ~ > 0 , ]~:C i,~,0 < ~ < I with i n space ~!~p(~) n w2~P(~), where 2 < p < 2 / ( 2 ~/~), D is a convex polygon in the plane, ~ is the largest angular value on the polygon's boundary. Finally~ recently in [ii], there has been established a Lipschitz estimate for divergence elliptic equations of second order in an arbitrary convex domain. The goal of the present paper is an extension of the result of [121 into the multidimensional case. Namely, we will investigate the behavior of a bounded generalized solution of the Dirichlet problem

7~i a~ (x, u, u~) = a (x, u, u~),

(])

x ~ G, ~ (z) = O, z ~ oG

(by reiterating indices, one means the summation from 1 to n) near a conical point ~ e ~G of a domain G c R ~ (n ~ 2) that is assumed to be smooth surface everywhere except for the origin ~. Our notation is widely accepted~ We assume that Eq. (i) is elliptic, and its coefficients satisfy the minimal smoothness condition and some growth restriction with respect to IvuI. We will prove that in some neighborhood of point

u(x)= O(Ix1~), Vu(x)= O ( I z t ~ ) with an exact value of % > 0 and that u(x) has second order generalized derivatives that are square sum~ab!e with some exact weight. In addition, we do not make any assumptions concerning the sign of u(x). i.

Notations,

Definitions,

Auxiliary Inegu_alities

We admit the following notation: Bd(0) - the ball in R ~ of radius d > 0 and center at the origin

~;

Go d = G N Bd(0) - the cone in R ~ with the vertex at ~ for some sufficiently small d > 0, i.e., Go d = {(r, ~)I0 < r < d; ~ = (~i, ~2 ..... ~n-1) e ~}; (r, m) - spherical coordinates of a point x e lRn; - a domain in the unit sphere S n-l with infinitely differentiable

boundary 3~;

L'vov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, NO. 6:~ ppo 25-38, November-December, 1990. Original article submitted June 14, 1988.

0037-4466/90/3106-0891512.50

9 1991 Plenum Publishing Corporation

891

F0 d = {(r, ~)I0 < r < d; m e 3fi} c a G

- the side surface of the cone

G~;

~p = G0dln {Ix I = p}, 0 < p < d. Recall some elementary formulas [13]:

(2)

dx = r~-Idr do~, df~ = p~-IdcD,

where dm is the area element of the surface of the unit sphere;

(3)

iv=l= (e?

where IVu I is the modulus of the gradient u(x), and IVmul is the projection of a vector Vu onto the plane that is tangent to the sphere S n-~ at the point w; Au = - -O~u + n - - t Ou + ~ A~u, Or2 r Or r~

(4)

where h is the Laplace operator in R '~, and h~ is the Beltrami-Laplace operator on the unit sphere. We introduce the following symbols for functional spaces: Lq(G) is the Banach space of measurable functions on G, which are summable to the power q ~ I, and which is endowed with the norm llUllq,G. Wk(G) is the Sobolev space of function from L2(G) possessing generalized o

derivatives up to the order k that are square summable on G. Wk(G) is the subspace of the space Wk(G), whose dense subset is the totality of all infinitely differentiable functions with supports in G. W01(G) is the subspace of the space WI(G), whose dense subset is the collection of all functions that are continuously differentiable on G and vanish on 8G. o

w a k ( G ) is the space of functions possessing generalized derivatives up to the order k in the interior of G and endowed with the norm k

I [14, 15], in particular

II

= 5

+

iv

I +

G n 2

U~xx

E

2

{,j=l

We will need certain inequalities. Ha~dy Inequality [16, p. 296, Theorem 330; 17, p. 238, Theorem V.IO].

For

every

u(r) e WI(]0; d[) d

d

y r=-~+=u 2 dr ~ ] 4 _ n _4 a I ~ j?r ~-a+=uTdr, 0

a<4--n,

(s)

0

provided the integral on the right hand side is finite. Generalized Wirtinger Inequality. problem

In the domain ~ c Sn-1 we consider an eigenvalue

A~u+~(~+n-2)u=O, for the Beltrami--Laplace tive eigenvalue X = t(~) we o b t a i n

892

o~,

u]ag=0

(6)

operator. From [13] i t i s known t h a t t h e r e e x i s t s t h e l e a s t p o s i related to this problem. With the help of the variation principle

fu 2dco ~< ~2 _~_~t(n -- 2) ! [ V ~ u t2 dco Remark.

The constants

in inequalities

(5), (7) are the best.

By multiplying inequality (7) by i/r and integrating by virtue of (2), (3), we obtain COROLLARY

i.

(7)

Vu ~ Wo1 (Q).

it with respect to r e

]0; d[, and

For every u e V = {v e Wi(G) Iv(x) = 0, x e F0al}

f.fr-nu2dx~;~2+

t

~ (n-- 2)

y.[ r~-" l V u 12dx

(8)

4 provided the integral on the right hand side is finite. Integrating COROLLARY

inequality

2.

(5) with respect to ~ e ~

we obtain

For every u e W01(G0d)

yyr~-4u~dx
(9)

4

provided the integral on the right hand side is finite. Generalized

Hardy-Wirtinger

Inequali~.

For every u e V = {v e Wi(G) Iv(x) = 0, x e

F0 d}

(i0) (~ ~<4 - -

n)

provided the integral on the right hand side is finite~ Proof. First, the inequality

exactly as in the proof of inequality

(8), from inequality

f y r~-~u2 dx <~ ~ § ~1(n_ 2) f 9lr~-tlV~u]2dx. ~do Odo If ~ < 4 - n, then inequality (I0) can be obtained by combining If a = 4 - n, then (i0) coincides with (8).

2.

,(7) we deduce

(il)

inequalities

(9) and (ii)~

A Weighted Energy Estimate

We will consider bounded generalized solutions L~(G) n W01(G), satisfying the integral identity

of problem (i), i.e. functions u(x) e

f ] [a~ (x, ~, u~) ~ (x) + a (x, ~, ~) ~ (x)] dx = 0 G

for

every

D(x) e w0i(G),

under

the

following

conditions:

(a) the functions ai(x , u, p), i = i . . . . . n are measurable for x e G and every u e R, p e R ~, differentiable with respect to pj~ j = i . . . . . n, and fulfilling inequalities

v([ul)~
Op~

j

-- ~ --~ K r o n e c k e r

symbol,

V ~ R ~, i, ] =

~ ....

(a~) ,

n,

(a2)

893

[

~=la~(x,u,p)]X/2~'~([ul)(Ipl

q>n, (b) a(x, equality

u,

p)

is

a measurable

+ g(x))

O
Lq(G),

g(0) < oo.

function

for

(a3)

x e G, u e R ,

p e R ~, s a t i s f y i n g

the

in-

la~(x, u, p)i ~<'~2~(1~I)((p2 + fi(x)), where 0 <_ f(x) ~ Lq/2(G), q > n, v(t)[~(t), ~1(t), ~2(t)] is a positive nonincreasing decreasing] function, defined for t _> 0, B, v~> 0, B~, ~2 _> 0.

[non-

Remark. Conditions (a), (b) guarantee [18, p. 519, Theorem 2.2] the Holder continuity in G of a bounded generalized solution of problem (i). In particular, computing the well known quantity M0 = v r a i m a x lu(x)l, one can find y > 0 and co > 0, depending on M0, n, q, ~, ~ , Ba, v, G , such that

!~t(x) 1 = I ~ ( x ) - e~(,0) I ~< c~lxl ~, IxI < d. THEOREM I. Let u(x) be a bounded generalized solution of problem (i). (a), (b), and the following condition be fulfilled:

Let assumptions

for every 6 > 0 there exists d o > 0 such that, for every p e R ~, there holds the inequality

( ~ [a~(x,u,p)

-

a~(O,O,p)]~) ~/~ <~5tPl + h(x),

-

(a~)

i=1

whenever

lxl + lu] < d o , 0 ~ h(x) e Lq, q > n.

If, additionally,

h(x) e Wa_2~

f(x) e Wa~

~ ! 4 - n, and

(12)

> 2 -~(n + ~)/2, then there holds the estimate

(13) G

where the constant c depends only on the quantities M0, g(0), mes fl, mes G.

v, Dl, ~2, D, a, n, I, q,

_

Proof.

Fix a point

For every e e

--

~ S ~ - n ~ L Q ' and consider the unit radius-vector

]0; d[, the vector el does not belong to Go d .

>

l=O~=(/I

.... , l~).

If r is the radius-vector

of a

point x e G, then the quantity r~ = It- eli ~ 0 u ~ G. Therefore the function D(x) = re~-2u(x), 0 < e < d/2 is admissible in the integral identity of problem (I), and the identity takes the form j ~

ai

=--

G

[

e

(x)a(x,u,u~)+(cz--2)u(x)r=-4a s

i

(x,u, ax)(x,

--

sl~)]dx.

(].4)

G

However, 1

a~(x, u,p) = p j )

~(~pj)

& + ai (z, u, 0).

(15)

o

In particular,

by virtue of assumption

(a2)

ai(0,0, p)=pi+ai, o ai (x,

894

a o__ i=ai

(0,0,0),

o u, p) p~ = I P Is + a~pi + [ai (x, u, p)

~---t, .... n,

(16)

- - a~ (0, 0, p)] pi.

(17)

Let us rewrite inequality (14), taking into account (16), (17) and the fact that integrals with multipliers ai ~ are equal to zero (by virtue of the Green formula and the boundary condition). Using the integration by parts for the integral containing the expression U-Uxi , we obtain

3 ~

IVu

dx=(2--

--

G

G

(18)

G

o ~m b ~~- ~ ~ + (a - 2) (x~ - sIi)r~-~u(x)]dx - o[ ~ r~-'u(x)a(x, u, u~)dx,

~r

G

Let d > 0 be some small number, to be selected later.

Let us split the domain G into two

subdomains Go d , G\G0 d and use the additivity of the integral. in (18) from the right~

We will estimate the integral

In t h e domain G\Go d ( r ~ d) we h a v e r ~ = ~ ( x ~ - - e l ~ ) 2 = r ~ - - 2 e ( x ~ l ~ ) + e ~ i ~ ( r - - e ) ~ > . ~ Therefore, ~=:

t:

9

r ~-~ s t t ~ d z ~ M~

GNGdo

@-)

~-~

e~

]

A[

V e ~ 0; 2L. "

(19) rues G .

By virtue of condition (a3) , in the domain G\G0 d there holds the inequality

( ~ : [=~ (., ,., =~) - ~ (o, o, u~)] 0 */' < ~: (Mo, ~:, g (o)) (~ + I v ~ I + g(x)),

(20)

and therefore, using HSlder~s inequality, we obtain

Y5 [ai(x'u'u~)--a:(O'O'u~)][r~-'uxi + (~ -- 2) r~-4~ (x)@~ -- ~h)] dx'--'--< G\Gdo

g2(x) + l)dx~c~(Mo, v, ~, ~:,g(O),

~.~c~ S ~ (I Vul 2 + G

(2l)

d,a, rues G) x ( t + i!g!lq,~ + ii /I',q,'~,& i'

by virtue of estimate [18, Chap. IX, Sec. 2, Lemma 2.1]

J'Iw

I= & ~< ~ (Mo, ~,,~, m ~ @ (~ + I!g I!~,~+ II/b/~,~)"

(22)

G

Finally, by assumption (b) and inequality (22), we have

S S r~s-2u(x) a (x, u, us) dx ~ ca (t

+ I[g l},e + []] l[q/2,~)

(23)

G\Gdo with a constant c 4 depending only on M0, ~, ~, ~2, ~, d, mesG~ in view of estimates (19)-(23), results the inequality

Thus, from inequality (18)~

~ jCr~-2 s I V u ] ~ d x < ~ ( 2 - - ~ ) ( 2 _ ! _ ~ ) f j'r~-~u2dx + %~ SY

[ai(x,u, ux)--ai(O,O,u~)] 2

~o~

:/=[r~-2lVul+(2--
(26)

4

with a constant c s depending only on M0, v, ~, ~i, P2, g(O), d, =, n, mes G.

895

Consider the domain Go d . In order to estimate the first integral on the right hand side of (24), we carry out a change of variables

y~x~--eli,

i = 1 , ..., n.

(25)

L e t ~r be t h e image o f t h e domain Go d u n d e r t h e c o o r d i n a t e t r a n s f o r m a t i o n main @r we a p p l y t h e g e n e r a l i z e d H a r d y - W i r t e n g e r i n e q u a l i t y ( 1 0 ) :

(25).

In the do-

y r~-au2(x>dx = ~ S r~-au'(y + sl)dy< (26)

H(s

f ~r~-21V,u(g + el)t2dy= H(~,,n,a) j j ~ IVu(x)

dx,

where

H(~, n, a) = [(2 -- (n + a)/2) z + ~(~ + n -- 2)] -~

(27)

is the constant in inequality (i0). Further, u(x) is Holder-continuous; therefore in Ga d holds the inequality Ix] + lu(x)l 5 d + c0dT. Moreover, assumption (a4) is satisfied. We choose a number d > 0 in such a way that, for a 6 > 0 given below, the following inequality would be valid

(2S)

d + cod ~ < do(~).

By assumption (a4), using the Cauchy inequality in (26), we obtain

y (la~ (x, u, ~,) - a~(0, o, ~)]~)'/" [~-~ I v~ I + (2 -~),-F~ I ,~ I] dx < (29)

5

+(2--a)H(L,n,a)

, r~-~lVu[2dx + - ~ . God

~

(x) dx V S > 0 .

Gao

Turning into the last integral in (24), by a Holder-continuity of u(x), by virtue of the Cauchy inequality which is satisfied for any o > 0, and by (24), we have

l Gd

l(Ivul

+/(x))

+

9

lw

+

d

d

Go

dx

W>0.

(30)

GO

Returning to inequality (24), by virtue of inequalities (26), (29), and (30), we infer that

d %

d GO

(31)

P'2(M~ f . I rc~/2(x)dx -I- 3~ -- c~j" J~ rsr162 , ~ , / ~dx +

where, by inequality (27) and condition (12),

C(~,n,a)= i - - ( 2 - - ~ ) ( 2 - - - ~ ) H ( ~ , n , a ) > O .

896

(32)

Since h(x) e W~_2~

f(x) e We~ 93

e.~JrO

then there exists

~ ~ r~

(x)dx = C~ r~-~h~ (x)dx,

llmoS!r~]2(x)dx=~!ref(x)dx"

(33)

The numbers 6 and o are chosen as follows: (%'2) ~2 (Mo) H (~, n, ~) = 6 [(5 - - ~)t2 + (2 - - ~)//(~,, ~, ~)] = C (~, n, ~)/0,

(34)

while the number d is selected in such a way that inequality (28) with 6, taken from (34), and the inequality

~(Mo)~od T~ C(%, n, ~)i6, hold.

(35)

Then inequality (31) takes the form

S .I r2-:l W l ~ * d Go

~ ~o.f J [,-<~s~(*) § ,.<~-~h:(,)] d~ § ~(~ + Ilu!I~,~ + I!!i!~/~.~) § ~ v • G

(36)

if 0 < E ! s0(<) [by (33)]; here the constant c 6 depends on X, n, a, M0, ~2, and c 7 depends on the same quantities and also on v, ~, DI, g(0), mes G, q, mes ~~ Since the right hand side of inequality (36) does not depend on ~, then, carrying the limit transition for s + +0, taking into account the arbitrariness of K in (33), and also bringing inequality (22) to our attention, we obtain estimate (13). Remark. Let us discuss condition (12) of Theorem 1 for n = 2. In this case G e ~G is a corner point, ~ = ]0, m0[, m0 is an angular value in the neighborhood ~ ; Go d = ]0, d[ x ]0, n0[. The eigenvalue problem (6) is the boundary Stourm-Louville problem e" + % 2 u = 0 , u = ~ ( ~ ) , ~-~Q,

~(0)

=

u (o~o)= O.

The least positive eigenvalue of this problem is equal to X = v/w0; condition (12) takes the form 2~/~ 0 > 2 - a; a S 2. Condition (12) and its accuracy has been established earlier in the papers [15, 19] in the case when Eq. (i) is linear. 3.

An Estimate of the Modulus of the Generalized Solution We begin by proving

LEMMA. Let u(x) be a generalized bounded solution of problem (i) and assume that assumptions (a), (b) are fulfilled. Then, for every function v(x) e V = {v e WZ(G0P)Iv(x) = 0~ x e FOP}, and almost all p e ]0, d[ there holds the inequality

G2

~P

(37)

Proof. Let Xp(x) be the characteristic function of the set G0 p, and (X0)h be its Sobolev a ~ . In the integral identity of problem (i) we set q(x) = v(x) (Xn)h(X) Vv e W0-(G), and hence the function D(x) is admissible by virtue of Theorem i, for a = 4 ~- n. In the obtained equality we carry the limit transition for h + 0, which results in obtaining the soughtfor inequality (37). The possibility of the limit transition is substantiated by methods of the theory of functions of real variable, combined with properties of functions' means [13, 17] and also with the already proven Theorem i. In particular, we employ Theorem ilI.lO [17, p. 113] and Lemma 1.3.8 [20, p. 37]. THEOREM 2. Let u(x) be a generalized bounded solution of problem (i). filled are assumptions (a), (b), and the condition

Assume that ful-

897

for every x e God, u = u(x), p e R ~ there holds the inequality

(~=: [a~(x,u, p)--a~(O, 0, p)12)]-/2< 8(,X [)[p [ + h(x),

(as) d

where 6(r) is a nondecreasing positive function satisfying Dini's condition y ~ d r < o o . 0

F u r t h e r , assume t h a t f u l f i l l e d (c)

are assumptions

a~(x, u, p)pi>~volpl2--~t31ul~--u2q~(x), u(x)a(x, u, p)~< ~o[pl 2 "F ~tal~l~'4- u2q~:(x), 2n/ ( n - - 2 ) > [ 5 > 2 ; O~q)(x)~Lq/2(G), q > n , vo>0; /~o, ~ta~0; o

0

o

(d) h.(x)~W2_,(G),

9

/(x)~WO_n(G) ' 92Ih~(p,o))dc~ § n v~o

s>2L(_Q),

O<9
Then the following estimate is valid

l~(x) l ~ clxl~(%

(38)

where l(~) is the least positive eigenvalue of problem (6); the constant c depends only on diam

n, q, rues g~, mes G, iJgilq,~, flfllq/2~, llq~tl~/2G, IIh[l~o, IIlllo o "

'

'

2--n

,

G

~ r(r___))dr. Mo, v, ~t, ~1, ~t2, ~t3, vo, ]to, ~, k, s,

W4--n(G) o

Proof. In identity (37) we set v(x) = r2-nu(x). Such a function is admissible by virtue of Theorem 1 and inequality (8). As a result, we obtain

~o

~o ~p

GP

We rewrite this equality, taking into account (16), (17). The integrals with multipliers ai ~ or those with expressions UUxi can be transformed by the Green formula:

coo

::

(39)

~o S ~ r2-~'u (x) a (x, u, u~) dx + p .[ u (x) [a~ (x, u, ux) --a~ (0, O, ux)] cos (r, xl)[ra~p. cpo Denote P

(4o) GPO

[by virtue of (2), (3)].

898

Then

0

(41)

v'(p)=S (pu~,+ FI V.ul~) do~.

We will estimate from above the integrals on the right hand side of (39). By the Cauchy inequality with an arbitrary ~ > 0, we have puup ! (e/2)u ~ + (i/2e)p=up 2 Apply Virtinger inequality (7) and put s = X ~ X(~):

n--2 2

5 u~doJ_}_ p Su u o d ~ o ~ e @ n2- - 2 Yuedco+ -2e.:r u o deo< ~~ d(p).

(42)

By virtue of inequality (as), the Cauchy inequality with ps ge > 0, and inequality (8)

~ [al (x, u, u:~) - - a~ (0, O, u~)] [r2-~u=~ -p- (2 -- n) r-'ixiu (x)} dx ~.~

4 5(p)+pe[ 2 -n-t-

s

n--2

(43)

n i p__~f --7-r~-'%2(x)dz V a > O .

] , v(p)

4 Analogously, using

(7)

instead of ( 8 ) ,

we infer that

P .I' u (x) [al (x, u, uX)" ai (o; o, u~)] cos (r, xO tr=od~ < (44) c(n, %)[5(p)@ p~]pv' (p) + t p~._~h~(p, co)&o W > O . .q

Finally, taking into account assumption (b) and Holder-continuity of u(x), we estimate the integral

] r~-nu (x) a (x, u, ux) dx <

copVp2 (Mo)

GP O

u I I (z) dx

~ .[ r27niVu 12dx@ GPO

<~~<.(Mo) , rL
j

'::

(45)

'

2 (~.~+ ( ~ - 2) X) u (p) @ ~- ~<' ('u~ p

- SSG~ r i - ' / e

(x) dx

Ve>O.

In inequalities (43)-(45) we will assume 0 < s 5 7, so that 07 ! p~. Returning to inequality (39), by virtue of estimates (42)-(45), and taking into account the notation in (40), (41), we have

..> 2% t -- ~ (p) Y(p) v' (P) ~ 7 i + o (,o)

2s

(n, Mo) ~-~-e, t + ~ (p)

(46)

where o(p) = c(n, X, M0) [6(0) + pe], 0 < e _< y, 0 < p < d. It is known that the solution of the differential equation (46) is majorized by a solution of the Cauchy problem for the ordinary differential equation

w'(p) -- 2~ i -- o (p) w(p) p i+o(p)

2~kcl -- -i-V~(p) pS 1 ~,

O
~v(d) -- %,

where

(13),

uo for

§ ~

~

~

i

n I

Writing

do~

the

"+[;

w(p)=uo~)

exp 2)~

p

+ tl/'ilq/ ' § llhll~176 ~ o lut

ion

0 (r)

dr

t+~(r)

T

of

]

this

-i-'ll/il;o"4-n,l by

problem

r;

of i n e q u a l i t y

-

'

+2k~,clp2~exp 2X ~-

virtue

p

g (r)

t+o(r)

dr

r

]

X

899

[r ~-l-e-2~"

J,7~-~(-~)

a(t)

exp --2h

0

dt

t+a(t)

t

dr

Ve~]O;~/],

O
r

we obtain v ( p ) ~ c 9 2x,

O
(47)

where the constant c depends only on quantities Mo, d, ~, ~t~, ~t2, ~t, n, )~, q , , m e s Q , mes G, ilgliq,~, d

l hli#0s_n,iO),ltilIwL~)' '

'o

' ttll!gl~,~,

u[6(r)dr, r

k,s.

On

stratum

the

Q'

=

{x'll/2

< I x ' i < 1)

consider

a

o

function

z(x') = p - ~ ( p x ' ) , taking

u m 0 outside

G.

This

function

is

O
a generalized

a~(x% z, z~,) ~ t)~ a~ (px', f~z, p

(48)

solution

i n Q' o f t h e

equation

(1')

z~,),

a(z', z, z=<,)~ p~-~a (px', p~z,:p~-h z~,),, By the embedding theorem [18, Chapter II, inequality (2.22)]

[ ~ l z (x')I~dz'<~ c{_[-Q;.[ (I V 'z 12-4-zD dx' ~q/', 2 ~.~ ~ 2nl(n- 2). ~r \ 2 Changing the variable x = px' in this inequality and substituting (48), and by virtue of inequality (47), we obtain the inequality

~y

q lu(x)idx<~cp

n+q~,

, 2<~q~.~2n/(n--2), n > 2 ,

p/2
'

which, for the function z(x'), takes the form

.f I z (x')iqdx ' <~ c', Q,

2 <~ q <~ 2n;'(n- 2), n > 2.

(49)

Equation (i') lies in the area of applicability of Theorem 7.6 [18, Chap IV]: indeed, the assumptions of our theorem and estimate (49) guarantee fulfillment of hypotheses of the indicated theorem. Thus, we obtain the estimate

:vrai max lz (x') I ~< M'o Q, w i t h a constantM0' depending only on quantities c' from (49), vQ, P0, N, D3, ~,

(50)

ilP2+X~(PX')liq/2,Q'.

Returning to the function u(x), from (50) and (48) we deduce Ju(x)J i M0pX(~), x 9 Go d n {p/2 < JxJ < p < d}. Putting Jxj = (2/3)p, we come to the sought-for estimate (38). This proves Theorem 2. 4.

An Estimate of the Second Order Derivatives of a Solution

THEOREM 3. Let u(x) be a generalized bounded solution of problem (i) and let assumptions of Theorem 1 be fulfilled. Assume additionally that, for x 9 G and every u(x), p 9 R ~ , the functions ai(x, u, p) (i = i, ..., n) and a(x, u, p) are differentiable with respect to their arguments, and there hold the inequalities

900

~=,%.~e~, ~ 71- ~

.

,+

~ i

(-

§ =

)

where

<'":

~p~(x),

1

<~.4(lul)(IPI + %(@, '

i = 0, i, 2 are nonnegative

functions,

and ~0(x), ~p2(x)~ Lq/~(G), (p~(x)~ Lq(G), q > n. o

Then u ( x )

e Wa=(G) and t h e r e

holds

the estimate

II ull~a(c ) < c,(! + Ilglb,G + [1] llq/=,~q- []% II,~/=,G-A-]1% ]~a/2,G+ IIqolb.~ + (51)

::/~(x) + :(x)]dx}

"~4/q

,

u<~4-n,

provided the latter integral is finite, the constant c I > 0 depends on the same parameters as in (13), and c a > 0 depends only on M0, v, p, v0, Pl, P2, Pd, Ps. Proof.

Consider a sequence of domains Gk, p = Go d n {p/2 k+1 < r < p/2k}, k = 0, i, .... so

0 < p < d.

Obviously,

G~.pcG~,

O Gh.p=G~ ,

In the domain Gk, p we consider Eq. (I) and per-

h=o

form a transformation of coordinates x i = xi'/2k-i ~ i = 1 . . . . . n. Under such transformation, the domain Gk, p is mapped into the domain Gz, p of the space of points x' elR h. The function v(x') = u(x'/2 k-l) is a bounded generalized d

':h),

solution of the equation

,

-77ai (x , v, Ux,) + a (k)(x',~v,vx,)=0, ax i

~i-(k~"(x', v, v~,) ------21-ka~ (21-kx ', v (x'), 2~-1t,'~,),

a(h)(x ', u, V,,)==2~(1--k)a(2*-~X', U(X'), 2h--'V~,)

x'~Gl,p, i -- 1, . .., n,

(Ik')

(k = O, l, 2 . . . . . ).

To equation (ik') we apply the well known results [18, 21] on smoothness of a generalized solution inside the domain and near a smooth piece of the boundary. Namely, from the proof of Theorem 1 [21] the validity of the following estimate is easily seen:

Gl,p

P

P

P

GoUG1UG2

P

,P, P

(Go UGI [JG2 ,~4lq

q- iq/2(x ') ~, q:~ (x') -.}- gq(x')]/z' } ,

q > n,

) where ci, c2 depend only on M0, v, p, Pl, P2, Pd, Ps. Reverting in this inequality to the variable x, by virtue of the definition of the sets Gk, p we obtain the inequality

Gh,p

Gk--l,pUGk,pUOh+ l,p

(52) [ h--l,pUCh,pUah+l,p

Let us sum up all equations (52) over k = I, 2 . . . . . The inequality obtained this way, combined with Theorem i and a result from paper [21], means that u(x) e W~2(G) and estimate (51) is valid. This completes the proof of Theorem 3.

901

(12).

COROLLARY. Let assumptions of Theorem 3 be fulfilled with the exceptions of condition The generalized solution of problem (i) u(x) ~ W2(G), if (i) n ~ 4, (ii) n = 2 and 0 < m 0 < ~ (cf. the remark on page 898),

(iii) n ~ 3 , Q c ~ 0 = { ~ = ( 0 ; ~)10
where ~0 is a solution of the

Proof. (i) By Theorem 3, u(x) e W4_n2(G). Condition (12) becomes trivial for a = 4 n because I = ~(~) > 0. The assertion follows from the inequality

and by v i r t u e

of (51).

( i i ) I n Theorem 3 we s e t a = 0. C o n d i t i o n ( 1 2 ) becomes a t r i v i a l n = 2, t h e a s s e r t i o n f o l l o w s f r o m t h e r e m a r k on p a g e 898.

inequality

(iii) Condition (12) is transformed into the inequality X(~) > 1/2. domain in which the eigenvalue problem (6) is solvable, for ~(~) = 1/2: h~u § (1/2)(1 + 1/2) u = o,

~

again.

For

Let ~0 c S 2 be a

~2o,

(53)

u I~o = o.

Then the condition X > 1/2 means that S c ~0 [7, p. 781, Proposition 2.1.1]. We will turn into problem (53) and will be looking for a partial solution in the form u ~ v(8). The function v(8) is subject to a boundary value problem for the ordinary differential equation

i

d sinO~- ~-

sin 0 dO

t-F

v-~O,

(54)

O
(55)

v ( - ~ 0 ) = v ( ~ o ) = 0.

The solution of Eq. (54) is the Legendre function of the first kind v(8) = P1/2(cos 8) which, on the interval 0 < e < ~, has exactly one zero (we denote it by mo) [22, p. 158, Example 39] (see also [14, p. 291] or [i, p. 52]). The corollary has been proved. 5.

An Estimate of the Modulus of the Gradient of a Solution

Theorem 4. Let u(x) be a generalized bounded solution of problem (i). Let the functions ai(x, u, p), a(x, u, p) be differentiable with respect to their arguments and fulfill conditions (a), (b), (e) with q = ~. Let all assumptions of Theorem 2 be fulfilled. Then there holds the estimate v a ( x ) t <~ clxl ~('~-~,

(56)

where l(~) is the least positive eigenvalue of problem (6); the constant c depends only on q u a n t i t i e s Mo, ~, ~o, ~, ~o, ~,, ~t2, ~ , ~t4, ~ , ~ , k, s, n, q, mesQ, mesG, diamG, Uh~l~o ,6)' v r a i m a x { l % ( x ) l ' 2--n~

~C--G

diamG

Ir

I,jI~2(x)I, Ig(x) l, I](x) l}' and a l s o on t h e v a l u e o f t h e i n t e g r a l

S ~-~dr. o

Proof. Consider in the domain Q' the generalized solution z(x' ) of equation (i') (see the proof of Theorem 2). The assumptions of our theorem guarantee fulfillment of hypotheses of Theorem 2 [21] on the boundedness of the modulus of the gradient of a solution inside the domain and near a smooth piece of the boundary:

vrai max [ V 'z [ ~< M'I,

Q,

902

where M I' > 0 is defined with the help of quantities v, v0, ~, ~z, D2, ~4, ~s, and also of vraimax Iz(x')l, which can be estimated using inequality (50). Returning to the function

Q,

u(x), according to (48), we will obtain

I v u (~)1<~ M~P x(u)-', x ~ God n {p/2 , < i x I <: I~ <: d}. Putting Ixl = (2/3)p, we arrive at the sought-for estimate (56).

Theorem 4 has been proved~

LITERATURE CITED .

2.

3.

4. 5 6 7 8 9 I0 Ii

12.

13.

V. A. Kondrat'ev and O. A. Oleinik, "Boundary value problems for partial differential equations in nonsmooth domains," Usp. Mat. Nauk, 38, No. 2, 4-76 (1983). V. N. Maslennikova and M. E. Bogovskii, "Lp-Theory of elliptic boundary-value problems for noncompact, nonsmooth boundaries," Funkcional~nye i chisiennye metody matematicheskoi Fiziki. Sbor. Nauch. Trud. Kiev, Naukova Dumka, 142-150 (1988); Rend~ del Semination Mat. e Fis. di Milano, 56, 125-138 (1986). E. Miersemann, "Zur Regularitat verallgemainerter Losungen yon quasilinearen elliptischen Differentialgleichungen zweiter Ordnung in Gebieten mit Ecken," Z. Anal. Anwend., i, No. 4, 59-71 (1982). E. Miersemann, "Quasilineare elliptische Differentialgleichungen zweiter Ordnung in mehrdimensionalen Gebieten mit Kegelspitzen," Z. Anal. Anwend., ~, No. 4, 361-365 (1983). E. Miersemann, "Boundary-value problems for quasiiinear elliptic equations of second order in domains with corners," Seminare Analysis !83-84, Berlin, 184-195 (1984). P. Tolksdorf, "On quasilinear boundary-value problems in domains with corners," Nonlinear Anal., ~, No. 7, 721-735 (1981). P. Tolksdorf, "On the Dirichlet problem for quasilinear equations in domains with conical boundary points," Commun. Par. Different~ Equat., 8, No. 7, 773-817 (1983)o P. Tolksdorf, "On the behaviour near the boundary of solutions of quasilinear equations," Analysis Munchen, 3, Nos. 1-4, 55-78 (1983). P. Tolksdorf, "Invariance properties and special structures near conical boundary points," Lect. Notes Math., 1121, 308-318 (1985). P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Programs Boston, London, Melbourne (1985). O. A. Ladyzhenskaya and N. N. Ural~tseva, "The Lipschitz estimate at boundary points for solutions of quasilinear equations of divergence type," Sib. Mat. Zh., 28, No. 4, 145-153 (1987). V. A. Kondrat'ev and M. V. Borsuk, "The behavior of the solution of the Dirichlet problem for a quasilinear elliptic equation of second order near a corner point," Differents. Uravn., 24, No. i0, 1778-1784 (1988). S. G. Mikhlin, Linear Second Order Equations [in Russian], Vysshaya Shkola, Moscow

(1977). 14. 15.

16. 17. 18. 19.

20. 21.

V. A. Kondrat'ev, "Boundary-value problems for elliptic equations in domains with conical or corner points," Tr. Mosk. Mat~ 16, 209-292 (1967). V. A. Kondrat'ev, "Smoothness of the solution of the Dirich!et problem for the elliptic equation of second order in a piecewise linear domain," Differents. Uravn., 6, No. i0, 1831-1843 (1970). G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge (1934). S. L. Sobolev, Introduction to the Theory of Cubature Formulas [in Russian], Nauka, Moscow (1974). O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1973). V. A. Kondrat'ev, I. Kopachek, and O. L. Oleinik, "Behavior of generalized solutions of elliptic equations of second order and systems of the elasticity theory in a neighborhood of a boundary point," Trudy Seminara Petrovskogo, 8, 135-152 (1982); Usp. Mat. Nauk, 36, No. i, 211-212 (1981). S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York (1975). Yu. N. Voinov, "Boundedness of gradients of generalized solutions of boundary-value problems for quasilinear elliptic equations near a boundary," Vestn. Leningr. Gos. Univ., i, No. 2, 5-13 (1974).

903

22.

E.T. Whittaker and J. N. Watson, Modern Analysis, Part 2, 4th ed., Cambridge University Press, Cambridge (1934).

ALGEBRAIC AND THE GEOMETRIC COMPONENTS OF THE CHEBYSHEV RANK OF SYSTEMS OF VECTOR-VALUED FUNCTIONS A. L. Garkavi

UDC 517.51

This paper is devoted to questions regarding the dimension of the set of polynomials of best approximation with respect to a system of functions with values in a Banach space. Let T be the segment [a, b] of the real axis R, let X be a Banach space of dimension q ! ~ with norm II'II, let C(T, X) be the space of continuous functions ~(t): T + X with the norm II~IIC = maxll~(t)ll. For X = R , instead of C(T, R ) we shall write C r. Assume further that S={~1(t), ., ~(t)} be a system of N linearly independent (LI) vector-valued functions ~ e C(T, X) (N < ~), let L(S) be the space of polynomials {p(t)} with respect to the system S. The number n(S) = N = dimL(S) is called the order of the system S. By A(S, ~) we denote the subset of the polynomials from L(S), providing Mithe best approximation E(S, ~)=in~{ll~'pi]c: p ~L(~)). As it is known, by the Chebyshev rank p(S) of the system S we mean the maximum of the dimensions of the subsets A(S, ~) over all ~ e C(T, X). The case p(S) = 0 corresponds to a unique polynomial of best approximation for all ~ e C(T, X). This case, under the assumption of the strict convexity of the space X, has been investigated in [i, 2] and in a series of more recent investigations of other authors. The idea of these papers goes back to the classical theorem of Haar [3], related to the space C r. In [4] one has established a criterion for a system S c C r of a prescribed Chebyshev rank. Generalizing, on one hand, the results of [2] and, on the other hand, the theorem in [4], V. A. Koshcheev [5] has obtained a characterization of systems S having a given Chebyshev rank in the space C(T, X) (for dimX < ~). In this paper the space X has been assumed to be also strictly convex (i.e. strictly normed). The characterizing conditions, established in the mentioned investigations (just as the conditions of A. Haar and G. Sh. Rubinshtein), have a purely algebraic character: they are formulated in terms of the linear structure of the space X and are not connected with its geometry (norm). In the case of a nonstrictly convex space, these conditions determine only a lower estimate Pa(S) for the Chebyshev rank p(S). Thus, for an arbitrary space X we have p(S) = Pa(S) + pg(S), where pa(S) is a lower estimate for p(S), due to the algebraic properties of the system S, while p~(S) ~ p(S) - pa(S) is the "increment" of the Chebyshev rank due to the geometric properties of the space X. For a strictly convex X we have p(S) = pa(S) and pg(S) = O. It is natural to consider pa(S) and pg(S) as the algebraic and the geometric components of the Chebyshev rank of the system S c C(T, X). We Shall call Pa(S) the algebraic rank of the system S, while ra(S) m n(S) is the (algebraic) corank of the system S. In this paper we investigate the effect of the geometric properties of the space X upon the Chebyshev rank of the system S c C(T, X). More precisely, the purpose of this paper consists in the determination of the maximum R(X, N, Pa) of the Chebyshev ranks p(S) in the class of all systems S c C(T, X) of a prescribed order n(S) = N and of a given algebraic rank Pa (S) = Pa, i.e., in the determination of the quantity R(X, N, Pa) = max{p(S): S c C(T, X), n(S) = N, Pa(S) = Pa}" A space X is said to be h-strictly convex if the supremum of the dimensions of the convex subsets of its unit sphere llxU = I is equal to h (see [6]). Obviously, h ! q - I, while h = 0 corresponds to a strictly convex norm. From q, n(S), and ra(S) one determines uniquely integers k, s m, d such that n(S) = (k - l)q + m, ra(S) = (s - l)q + d, 0 ~ m, d ~ q - i, ~ k. Consequently, Pa(S) = n(S) - ra(S) = (k - s + m - d (for q = ~ we take k = ~ = i).

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 6, pp. 39-45, November-December, 1990. Original article submitted August 23, 1988.

904

0037-4466/90/3106-0904512.50 9 1991 Plenum Publishing Corporation