A Boundary Value Problem for a Hyperbolic Equation with a Third-Order Wave Operator

Differential Equations, Vol. 40, No. 2, 2004, pp. 218–226. Translated from Differentsial’nye Uravneniya, Vol. 40, No. 2, 2004, pp. 208–215. c 2004 by Korzyuk. Original Russian Text Copyright

PARTIAL DIFFERENTIAL EQUATIONS

A Boundary Value Problem for a Hyperbolic Equation with a Third-Order Wave Operator V. I. Korzyuk Institute for Mathematics, National Academy of Sciences, Minsk, Belarus Received July 10, 2001

This paper is actually a continuation of [1], where the energy inequality kukE ≤ ckAukL2 (Ω)

(1)

was proved for a boundary value problem for the third-order linear hyperbolic equation Au = A1 u + A2 u = f (x),

(2)



 ∂2u ∂ n 2 x = (x1 , . . . , xn ) ∈ Ω ⊂ R , A1 u = − a ∆u , ∂x1 ∂x21   X ∂ ∂ α A2 u = , aα (x)D u, D= ,..., ∂x1 ∂xn

∆=

n X i=2

∂2 , ∂x2i

|α|≤2

where α = (α1 , . . . , αn ) is a multi-index, |α| = α1 + · · · + αn , Ω is a bounded domain in Rn with piecewise smooth boundary ∂Ω, and L2 (Ω) is the space of square integrable functions in Ω. In the present paper, we use variable-step averaging operators to prove the existence and uniqueness theorem for a strong solution of the boundary value problem for Eq. (2). Let us state this problem using the notation in [1]. By ν(x) = (ν1 (x), . . . , νn (x)) we denote the unit normal on the hypersurface ∂Ω at a point x ∈ ∂Ω. Let δ be a sufficiently small positive number. We assume that the boundary ∂Ω can consist of hypersurfaces of seven kinds, which can be defined as follows in terms of δ and ν(x) :   n X 3 2 2 S0 = x ∈ ∂Ω A1 (ν) = ν1 − a ν1 νi > 0, ν1 > 0 , i=2

S1 S2 S3 S4 S5 S6

= {x ∈ ∂Ω = {x ∈ ∂Ω = {x ∈ ∂Ω = {x ∈ ∂Ω = {x ∈ ∂Ω = {x ∈ ∂Ω

| | | | | |

A1 (ν) = 0, ν1 > 0} , A1 (ν) ≤ −δ, ν1 ≥ δ} , A1 (ν) = 0, ν1 = 0} , A1 (ν) ≥ δ, ν1 ≤ −δ} , A1 (ν) = 0, ν1 < 0} , A1 (ν) < 0, ν1 < 0} .

Other hypersurfaces are excluded from ∂Ω. The simplest example is the deformed (stretched) ball of radius R shown in the figure. We consider this domain in a Cartesian coordinate system x1 , . . . , xn in which the center of the ball coincides with the origin. Here all above-mentioned parts of the hypersurfaces Si (i = 0, . . . , 6) of the boundary ∂Ω are present. They can be described with the use of the conical surfaces n o 2 2 Zm = x | x21 − (tan ϕm ) |x0 | = 0 (m = 0, 1, 2), 2

where 0 < ϕ0 < ϕ1 < ϕ2 < ϕ3 = π/2, tan ϕ0 = |a|, and |x0 | =

Pn i=2

x2i .

c 2004 MAIK “Nauka/Interperiodica” 0012-2661/04/4002-0218

A BOUNDARY VALUE PROBLEM FOR A HYPERBOLIC EQUATION

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Figure.

Here S0 is the upper part of the surface of the sphere |x| = R lying inside the cone n o 2 Z0 = x | x21 − a2 |x0 | = 0 ; i.e., S0 =

  n X x |x| = R, ν12 − a2 νi2 > 0, ν1 > 0 , i=2

where ν = (ν1 , . . . , νn ) is the unit outward normal vector. The surface S1 lies between the cones Z0 and Z1 . More precisely,   n X R 2 2 2 S 1 = x ν1 − a νi = 0, ν1 > 0; |x| = R for x ∈ Z0 , |x| = for x ∈ Z1 , cos (ϕ1 − ϕ0 ) i=2 S2 = {x | |x| = R/ cos (ϕ1 − ϕ0 ) , |x0 | cot ϕ2 < x1 < |x0 | cot ϕ1 } , S3 = {x | |x0 | = R cos (ϕ3 − ϕ2 ) / cos (ϕ1 − ϕ0 ) , − |x0 | cot ϕ2 ≤ x1 ≤ |x0 | cot ϕ2 } , S4 = {x | |x| = R cos (ϕ1 − ϕ0 ) , − |x0 | cot ϕ1 < x1 < − |x0 | cot ϕ2 } ,   n X R S5 = x ν12 − a2 νi2 = 0, ν1 < 0, |x| = R for x ∈ Z0 , |x| = for x ∈ Z1 , cos (ϕ1 − ϕ0 ) i=2 S6 = {x | |x| = R, − R ≤ x1 < − |x0 | /|a|} . The boundary value problem consists of Eq. (2) in Ω and the homogeneous boundary conditions 2 u ∂u ∂ = 0, u|S6i=2 Si = = (3) ∂ν S6 Si ∂ν 2 S6 i=4

where ∂/∂ν is the derivative along the normal ν and ∪ stands for the union. It is of interest to consider problem (2), (3) for n > 2, since such problems were already considered, e.g., in [3–5] in the case of two independent variables on the plane (n = 2) for hyperbolic equations and systems. The proof of the energy inequality (1) in [1] shows that, for n > 3, there are new essential difficulties in the study of boundary value problems for hyperbolic equations defined in noncylindrical domains. Therefore, the methods used for plane problems in [3–5] cannot be DIFFERENTIAL EQUATIONS

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directly applied to our case. If the energy inequality (1) is valid, then the variable-step averaging operators [2] permit one to show that problem (2), (3) is well posed and prove the existence of a strong solution under some conditions on the problem data. Similar results for second-order hyperbolic equations of rather general form can be found in [6]. The statement of problems and the proof of their solvability, especially for the case in which the main equation is defined in a noncylindrical domain, are topical for higher-order hyperbolic equations. We also note the paper [7], where boundary value problems were considered for a hyperbolic equation in divergent form for n = 3 in very complicated domains.  ¯ | x1 = t be a cross-section of Ω ¯ = Ω ∪ ∂Ω perpendicular to the x1 -axis. Let S(t) = x ∈ Ω
3 ¯ ¯ we introduce the norm For functions u ∈ C Ω (three times continuously differentiable in Ω), kukE = sup

X

¯ x∈Ω |α|≤2

kD α ukL2 (S(t)) ,

(4)

where k·kL2 (S(t)) is the norm of the space of
Lebesgue square integrable functions on S(t). By D (A) ¯ satisfying condition (3). The energy inequality (1) with we denote the set of functions u ∈ C 3 Ω constant c > 0 independent of u was proved in [1] for every function u ∈ D (A) and for the operator A under some conditions on the boundary ∂Ω. By E we denote the closure of D (A) in the norm (4). Then the operator A treated as a mapping ¯ This can be justified by verifying the of E into L2 (Ω) with domain D (A) admits the closure A. closability criterion: Auk → 0 in the norm of L2 (Ω) as k → ∞ whenever kuk kE → 0 as k → ∞ [6].
¯ = f with f ∈ L2 (Ω) Definition 1. A function u ∈ D A¯ satisfying the operator equation Au is called a strong solution of problem (2), (3). By passing to the limit in inequality (1), we obtain the
same energy inequality for the operator ¯ ¯ A with the same constant c for each element u ∈ D A . To prove the existence Pn of a strong solution, we introduce some more additional conditions on Ω. Let P (ξ) = ξ12 − a2 i=2 ξi2 = 0 be the characteristic cone for the wave operator in Ω. We define a vector field R of elements r(x) = (1, r2 (x), . . . , rn (x)) as follows. Let b(i) (x) (i = 1, 2) be ¯ and let b(i) (x) ≥ δ1 for some arbitrarily small but nonzero δ1 > 0. some positive functions in Ω, We assume that r(x) = b(1) (x)r (1) (x) + b(2) (x)r (2) (x), where the vectors r(x), r (i) (x) (i = 1, 2), and (1, 0, . . . , 0) lie in some two-dimensional plane γ(x) passing through the vertex of the characteristic cone and the r (i) (x) are perpendicular to the generators ξ (i) (x) obtained in the intersection of γ(x) with P (ξ) = 0. ¯ generates a The vector field R , whose elements are uniquely determined at each point x ∈ Ω, family {%} of lines to which it is tangent. Definition 2. A subdomain Qj ⊂ Ω is said to be line % to which R is tangent is connected.

R -convex

if the intersection of Qj with each

Condition 1. The domain Ω can be partitioned by cross-sections S(t) into finitely subdomains Qj (j = 1, . . . , j0 ) satisfying the following conditions: (i) for each j, there exists a vector field R such that Qj is R -convex;
¯ j , k = 2, . . . , n; (ii) rk (x) ∈ C 3 Q (iii) (r(x), ν(x)) = 0 for each x ∈ ((S2 ∪ S3 ∪ S4 ) ∩ ∂Qj ). Condition 2. Each subdomain Qj ⊂ Ω (j = 1, . . . , j0 ) is convex with respect to straight lines parallel to the x1 -axis. Theorem. Suppose that Conditions 1 and 2, as well as the assumptions of the theorem in [1] providing the validity of the energy inequality, be valid, S0 ∪ S1 6= ∅, S5 = ∅, and S6 6= ∅ (where ∅ is the empty set). Then for each function f ∈ L2 (Ω), there exists a unique strong solution of problem (2), (3). DIFFERENTIAL EQUATIONS

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Proof. Since the problem is linear, it follows that the uniqueness of a strong solution is an ¯ immediate consequence of the energy inequality for A. By the general theory of closable operators, to prove the existence of a strong solution of problem (2), (3) for every f ∈ L2 (Ω), it suffices [6] to prove that the range R(A) of A is dense in L2 (Ω). If we also use the method of continuation with respect to a parameter (see the proof of the theorem in [8]), then it suffices to prove that the set R (A1 ), where the domain D (A1 ) of the operator A1 coincides with D (A), is dense in L2 (Ω). Let v ∈ L2 (Ω) be a function such that the inequality (A1 u, v)L2 (Ω) = 0

(5)

o n
¯ | u satisfies condition (3) . Here is valid for all u ∈ D (A1 ) = u ∈ C 3 Ω  A1 =

∂2 − a2 ∆ ∂x21



∂ . ∂x1


¯ and satisfies the boundary If u ∈ D (A1 ), then the function w = ∂u/∂x1 belongs to the class C 2 Ω conditions ∂w S w| 6i=3 Si = = 0. (6) ∂ν S6 Relation (5) can be restated as (L w, v)L2 (Ω) = 0

(7)

o n
¯ | w satisfies condition (6) , where w ∈ C2 Ω

∂2 − a2 ∆ and ∂x21 v ∈ L2 (Ω). If one uses the variable-step averaging operators Eδ in which the partition of unity is such that the boundary conditions are preserved, then Eδ w ∈ D (L ) for each w ∈ D (L ). Here Eδ is a variable-step averaging operator [2], where the parameters δ = (δ0 , δ1 , . . . , δk , . . .) depend on w, δk > 0 (k = 0, 1, . . .), and δk → 0 as k → ∞. The partition of unity in the construction of Eδ is such that the subdomains accumulate near the boundary ∂Ω to the effect that the boundary conditions for w are preserved. Therefore, Eδ w ∈ D (L ) provided that w ∈ D (L ). For this choice of Eδ w, relation (7) can be rewritten in the form for all w ∈

D (L )

=

L

=

0 = (L Eδ w, v)L2 (Ω) = (w, L Eδ∗ w)L2 (Ω) + (L Eδ w − Eδ L w, v)L2 (Ω) ! Z n X ∂w ∂w 2 + ν1 − a νi Eδ∗ v ds ∂x1 ∂x i i=2 ∂Ω ! Z n X ∂Eδ∗ v ∂Eδ∗ v 2 − w ν1 − a νi ds, ∂x1 ∂xi i=2

(8)

∂Ω

where Eδ∗ is the adjoint of Eδ , for each w ∈ that relation (8) is valid provided that

S

Eδ∗ v| Consider the commutator

4 i=0

D (L ).

Si

By varying w within

D (L ),

one can show

∂ ∗ = E v = 0. ∂ν δ S0

(9)

L Eδ w − Eδ L w, which can be represented in detailed form as

(L Eδ w − Eδ L w) (x) = (K0 w) (x) +

n  X i=1

DIFFERENTIAL EQUATIONS

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∂w Ki ∂xi

 (x),

(10)

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KORZYUK

where

" # ∞ n 2 X X ∂ 2 ψk (x) ∂ ψ (x) k (K0 w) (x) = (Aδ w) (x) − a2 (Aδ w) (x) , 2 2 ∂x ∂x 1 i i=2 k=1     ∞ X ∂w ∂ψk (x) ∂w K1 (x) = Aδ (x), ∂x1 ∂x1 ∂x1 k=1     ∞ X ∂w ∂ψk (x) ∂w 2 Ki (x) = −a Aδ (x), i = 2, . . . , n, ∂xi ∂xi ∂xi k=1 ∞

{ψk (x)}k=1 is a partition of unity, and the Aδk are the Sobolev averaging operators. The operators Ki (i = 0, . . . , n) are also variable-step averaging operators preserving the boundary conditions on ∂Ω. By taking account of (9) and (10), one can represent relation (8) in the form

0 = (w, L

Eδ∗ v)L2 (Ω)

n X ∂ − Ki∗ v ∂x i i=1

w, K0∗ v

+

! +

n Z X i=1

L2 (Ω)

w (Ki∗ v) νi ds

(11)

∂Ω

for each w ∈ D (L ). The operator Ki∗ (i = 0, . . . , n) is the adjoint of Ki . It follows from (11) that Ki∗ v = 0

(12)

on S0 ∪ S1 , i = 0, . . . , n. Therefore, (w, L

Eδ∗ v)L2 (Ω)

+

w, K0∗ v

n X ∂ − Ki∗ v ∂x i i=1

! = 0.

(13)

L2 (Ω)

Since D (L ) is dense in L2 (Ω), it follows that relation (13) is valid for each w ∈ L2 (Ω). Let the subdomain Q1 ⊂ Ω (see Condition 1) lie “above” the other Qj (j = 2, . . . , j0 ); i.e., Sj0 ˜ 1 be ˜ 1 and Q1 \Q x1 > y1 for each x = (x1 , . . . , xn ) ∈ Q1 and each y = (y1 , . . . , yn ) ∈ j=2 Qj . Let Q
˜ 1. ˜ 1 lies “above” Q1 \Q the subdomains of Q1 obtained from Q1 by the cross-section S t˜ , where Q In (13), we set

  w(x) =

I Jv =

x

˜1 Eδ∗ v ds for x ∈ Q

(14) ˜ 1. for x ∈ Ω\Q 
 Hx ˜ 1 ∩ S6 to a point x belonging Here x˜ is a curvilinear integral from a point x ˜ ∈ S˜− = S t˜ ∪ ∂ Q ˜ 1 ∩ (S0 ∪ S1 ) along a line % to which the vector field R is tangent. to ∂ Q We substitute the function w given by (14) into (13) and transform relation with  the resulting  ∂ the use of the boundary conditions (9) and (12) and the relation Jv (x) = (Eδ∗ v) (x) for ∂r ˜ 1 . Then we obtain x∈Q 

∂ ∂2 Jv, Jv ∂x1 ∂x1 ∂r





x ˜

0

 n  X ∂ ∂2 −a Jv, Jv ∂xi ∂xi ∂r ˜1) ˜1) L 2 (Q L 2 (Q i=2   n X ∂ ∗ ∗ = (Jv, K0 v)L2 (Q˜ 1 ) + Jv, Ki v . ∂xi ˜1) L 2 (Q i=1 2

DIFFERENTIAL EQUATIONS

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Further, Z 

1 2

∂ Jv ∂x1

2

˜1 ∂Q

2 n Z  ∂ a2 X rν ds − Jv rν ds 2 i=2 ∂xi ˜1 ∂Q

=

(Jv, K0∗ v)L2 (Q˜ 1 )

 Z n  X ∂ ∗ + Jv, Ki v + ∂xi ˜1) L 2 (Q i=1

(16) (Jv)ds,

F

˜1 Q

where rν = ν1 +

Pn

νi ri , ν = (ν1 (x), . . . , νn (x)), r = (1, r2 (x), . . . , rn (x)), and ∂ quadratic form in the first derivatives Jv, i = 1, . . . , n. We have ∂xi i=2

X ∂ ∂ Jv = Eδ∗ v − ri (x) Jv. ∂x1 ∂x i i=2

F

(Jv) is a

n

(17)

It follows from (17) that

∂ x0 = (x2 , . . . , xn ) , Jv t˜, x0 = Eδ∗ v t˜, x0 , ∂x1 n X ∂ ∂ ˜1 ∩ Jv(x) = − ri (x) Jv(x), x ∈ ∂Q ∂x1 ∂x i i=2

(18) 4 [

! Si

.

(19)

i=0

˜ 1 ∩ (S3 ∪ S4 ∪ S6 ). By virtue of the definition of J, we have rν ≤ 0 and Jv(x) = 0 on S˜˜ − = ∂ Q Therefore, 2 2 Z  Z  n ∂ ∂ a2 X − Jv rν ds + Jv rν ds ≥ 0. (20) ∂x1 2 i=2 ∂xi ˜ ˜− S

˜ ˜− S

Relations (17)–(20), together with (16), imply the inequality Z S(t˜)

2 (Eδ∗ v)

Z n
0 X  2  ∂ ∂ 0 ˜ t, x dx + a δij − ri (x)rj (x) Jv Jvrν ds ∂x ∂x i j i,j=2 ˜+ S

 Z n  X ∂ ≤ 2 (Jv, K0∗ v)L2 (Q˜ 1 ) + Jv, Ki∗ v + F (Jv)dx ∂x i ˜ L2 (Q1 ) i=1 ˜1 Q !

2 n n X X

∂ 2 2

≤ c1 kEδ∗ vkL2 (Q˜ 1 ) + kJvk2L2 (Q˜ 1 ) + kKi∗ vkL2 (Q˜ 1 ) ,

∂xi Jv ˜ + L2 (Q1 ) i=2 i=0

(21)

˜ 1 ∩ (S0 ∪ S1 ), δij is the Kronecker delta, and c1 is a positive constant. where S˜+ = ∂ Q To apply
the Gronwall H x¯ ∗ inequality to (21), along with the function Jv(x), we introduce the ˜ function J v (x) = x Eδ v ds, where the integration is also performed along the lines % to which R is tangent and z ∈ S˜+ . It follows from the definition of J and J˜ that     (Jv)(x) + J˜v (x) = J˜v (˜ x) . (22) Note that the points on the integration line % are functionally related. In particular, the zi = zi (˜ x) − ˜ (i = 1, . . . , n) thus defined can be treated as continuous functions with domain D (zi ) = S and range R (zi ) = S˜+ . DIFFERENTIAL EQUATIONS

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˜ Then we obtain In inequality (21), we use relation (22) and replace the function Jv by Jv. Z n X






a δij − ri rj z(x) 2

i,j=2 ˜− S

 ∂ ˜ ∂ ˜ Jv (˜ x) Jv (˜ x) βi (˜ x) βj (˜ x) (rν ) (z (˜ x)) β (˜ x) ds ∂xi ∂xj

+ kEδ∗ vkL2 (S(t˜)) 2

2

2 ˜ ≤ c1 (ε0 ) kEδ∗ vkL2 (Q˜ 1 ) + J˜v (˜ x) − Jv(x)

(23)

˜1) L 2 (Q

!

2 n n X X

∂

∂ 2 ˜ x) − ˜

Jv(x) + kKi∗ vkL2 (Q˜ 1 ) ,

∂xi Jv (˜

˜ + ε0 ∂x i L2 (Q1 ) i=2 i=0 where the functions βi (˜ x) 6= 0 can be chosen on the basis of the vector field R , β (˜ x) ≥ c2 > 0, and c1 (ε0 ) is inversely proportional to ε0 > 0 as ε → 0. By the definition of the Pn vector field R , for each Pn vector ξ(x) = ¯(ξ1 (x), . . . , ξn (x)) orthogonal to r(x), we have ξ12 − a2 i=2 ξi2 ≤ −c3 |ξ|2 = −c3 i=1 ξi2 for all x ∈ Ω with some constant c3 > 0. As such a vector, we take ! n X ∂ ˜ ∂ ˜ ∂ ˜ ξ(˜ x) = − Jv (˜ x) , β2 (˜ Jv (˜ x) , . . . , βn (˜ Jv (˜ x) . ri (z (˜ x)) β (˜ x) x) x) ∂x ∂x ∂x i 2 n i=2 In particular, it is obviously orthogonal to the vector r (z (˜ x)) = (1, r2 (z (˜ x)) , . . . , rn (z (˜ x))) for − ˜ each x ˜ ∈ S . Since βi (˜ x) 6= 0 (i = 2, . . . , n), we have Z n X






a δij − ri rj z(x) 2

i,j=2 ˜− S

 ∂ ˜ ∂ ˜ Jv (˜ x) J v (˜ x) βi (˜ x) βj (˜ x) (rν ) (z (˜ x)) β (˜ x) ds ∂xi ∂xj

n X

∂

˜

≥ c4

∂xi Jv i=2

Further,

(24) ,

c4 > 0.

˜− ) L 2 (S

2 n X

∂

∂

˜ (˜ Jv x) − J˜v(x) +

˜ ˜1) ∂xi ∂xi L 2 (Q L2 (Q1 ) i=2 !

n   2 X ∂

2

˜1 ˜ ˜ x) ≤ ε1 Q +

Jv (25)

∂xi Jv (˜

˜− ˜− ) L 2 (S L 2 (S ) i=2 !

2 n

2 X

∂

˜

˜ + c5 J v , +

∂xi Jv ˜ ˜1) L 2 (Q L2 (Q1 ) i=2
˜ 1 can be chosen sufficiently small if we appropriately diminish the where the positive constant ε Q ˜ 1 . By the properties of averaging operators, we have domain Q

2

˜

˜ x) − Jv(x)

J v (˜

n X

kKi∗ vkL2 (Q˜ 1 ) ≤ c6 kvk2L2 (Q˜ 1 ) . 2

(26)

i=0

Along with (23), consider the inequality

2

˜

Jv

˜− ) L 2 (S

2

˜ 2 ≤ c7 kEδ∗ vkL2 (Q˜ 1 ) + Jv

˜1) L 2 (Q

,

DIFFERENTIAL EQUATIONS

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which is obtained from the relation Z Z h i 1  ˜ 2 ˜ (˜ Jv (˜ x) rν (z (˜ x)) β (˜ x) ds = Eδ∗ v(x) Jv x) − J˜v(x) dx; 2 ˜− S

˜1 Q

in turn, the last formula can be derived from the relation 1 ∂ (Jv)(x) = Eδ∗ v(x)Jv(x) 2 ∂r ˜ 1 . By adding inequalities (23) and (27), by taking account of (24)–(26), and by by integration over Q choosing an appropriate value of ε1 , we obtain the inequality # Z " 2 X n  ∂ 2 ˜ kEδ∗ v(x)kL2 (S(t˜)) + J˜v + Jv (˜ x) ds ∂x i i=2 ˜− S   (28) # Z " 2 X n  ∂   2 ˜ ˜ ≤ c8 (ε) kEδ∗ v(x)kL2 (S(t˜)) + Jv Jv (x)dx + ε0 c9 kvk2L2 (Q˜ 1 ) . + ∂x i i=2 ˜1 Q

It follows from (28) that tZmax




Φ t˜ ≤ c10 (ε0 )

Φ(t)dt + ε0 c11 kvkL2 (Q˜ 1 ) ,

(29)

t˜

where




˜ 2 Φ t˜ = kEδ∗ v(x)kL2 (S(t˜)) + Jv

n X

∂

˜

Jv +

∂xi L2 (S(t˜)) i=2

,

tmax = max x1 , ¯ x∈Ω

L2 (S(t˜))

with a positive constant c10 (ε0 ) inversely proportional to ε0 . Inequality (29) can be obtained in a τ τ ˜ ˜ ˜ ˜ similar way for any subdomain Q ⊂ Q1 ⊂ Ω, where Q = x ∈ Ω | t ≤ τ < x1 < tmax . Therefore, tZmax

Φ(τ ) ≤ c10 (ε0 )

Φ(t)dt + ε0 c11 kvkL2 (Q˜ τ ) ,

(30)

τ

where Φ(t) =

kEδ∗ vkL2 (S(τ ))



˜ + Jv



∂ ˜

Jv + ∂xi L2 (S(τ ))

. L2 (S(τ ))

The Gronwall inequality, together with (30), implies that Φ(τ ) ≤ ε0 c11 exp {c10 (ε0 ) (tmax − τ )} kvkL2 (Q˜ 1 ) , 
whence it follows that kEδ∗ vkL2 (S(τ )) ≤ ε0 c11 exp c10 (ε0 ) tmax − t˜ kvkL2 (Q˜ 1 ) . On the left-hand side of the last inequality, we pass to the least upper bound with respect to τ , where τ varies from t˜
2 2 ∗ to tmax . Further, obviously, kEδ vkL2 (Q˜ 1 ) ≤ tmax − t˜ supt˜<τ
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c11 exp c10 (ε0 ) tmax − t˜ kvkL2 (Q˜ 1 ) . 2004

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KORZYUK

By choosing appropriate values of ε0 and t˜, we make the coefficient occurring on the right-hand side less than 1, say, 1/2; then the estimate can be represented in the form kEδ∗ vkL2 (S(τ )) ≤ (1/2)kvkL2 (Q˜ 1 ) . P∞ By passing to the limit as |δ| = k=0 δk → 0 on the left-hand side in the last inequality, we obtain the equality kvkL2 (Q˜ 1 ) = 0. By continuing this procedure, in finitely many steps, we show that v = 0 in the entire subdomain Q1 ∈ Ω. By moving further downwards, in finitely many steps, we then show that v = 0 in the entire domain Ω. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

Korzyuk, V.I., Differents. Uravn., 1991, vol. 27, no. 6, pp. 1014–1022. Burenkov, V.I., Sobolev Spaces on Domains, Stuttgart, 1998. Thome, V., Math. Scand., 1955, vol. 3, pp. 115–123. Thome, V., Math. Scand., 1957, vol. 5, pp. 93–113. Thome, V., Math. Scand., 1958, vol. 6, no. 1, pp. 5–32. Korzyuk, V.I., Vestn. Bel. Univ. Ser. 1, 1996, no. 3, pp. 55–71. Korzyuk, V.I., Differents. Uravn., 1997, vol. 33, no. 12, pp. 1683–1690. Korzyuk, V.I., Differents. Uravn., 1970, vol. 6, no. 2, pp. 343–357. Dezin, A.A., Obshchie voprosy teorii granichnykh zadach (General Questions of the Theory of Boundary Value Problems), Moscow, 1980. 10. Sobolev, S.L., Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike (Some Applications of Functional Analysis in Mathematical Physics), Leningrad, 1950.

DIFFERENTIAL EQUATIONS

Vol. 40

No. 2

2004