< C »(ll/lf_1IM’1 +m°H4f'+ M f + M f - , + II«ollfxC0’al)

A N N A LE S SO C IE TA T IS M A TIIE M A TIC A E P O LO N AE Series I : COM M ENTATIO NES M A TH EM A T IC AE X V I I I (1975) R O C Z N IK I P O L SK IE G O T O W A R Z Y S T W A M AT EM A T Y CZNE G O

Séria I : P R A C E M ATE M A TY C ZN E X V I I I (1975)

H. Kolakowski (Warszawa)

Non-coercive m ixed problem

In recent years there appeared many papers concerning the non-coercive problems of the form

Lu — f in Q,

(1) 7

Bu\aa = g ,

where L is a differential operator of order m, elliptic in the bounded domain Ü <=. Bn. For such problems it is possible to estimate the norm INIe+m-e (<$ > 0) in the Sobolev space Hs+m_d(Q) by ||/||s and suitable norms of the functions g on the boundary dQ. From these estimates various facts concerning solvability of problem (1) follow. Problems for which such estimates do not hold have also been studied. Many results have been obtained by the method of the theory of pseudo-differential operators.

As far as we know, no papers dealing with mixed problems for hyper­

bolic equations with the boundary conditions of non-eoercive type have been published.

The theorem stated below should be treated as the first step in this direction.

In this paper we shall consider the following mixed problem Uu—L u = f ( x , t ) , te (Q, T), XeQ,

ихг |dflx[0,T] = t)i u\t=0 = %|i=o = with the additional condition

( 3 ) u \rx[0,T] = u o(M ) ?

where Q is a bounded domain in B n, such that 1° the boundary of Q is a smooth manifold,

2° Q is a convex set with respect to aq, i.e. every set {x = ( x1^{, x' ):}

xc Q } x' — const} is convex, 3

3 — R oczn ik i PTM — P ra ce M atem atyczn e X V IH

190 H. Kolakowski

3° vector field dfdxl is tangent to dQ along

Г = {ж: же d/2, = 0}, dim/ 1 = n — 2.

We assume

/ж = a{j(x, t)ux.x. \ Moo, t)ux. + b(x, t)u,

i,j i

L: C“ (Æ"+1)^C °” (iJ“+1),

( o > 0)

i9j i

for every (ж, tf)e Q x [0 , 17] and £е.йи\ {0}, coefficients of the operator L does not depend on x x. The method of investigation of problem (2)-(3) is based on the well-known facts from *the theory of well posed problems for hyperbolic equations (cf. [1]).

Let / j ( û x [ 0 , î ] ) be a subspace of the space HS(Q x [0, T]) ([2]) with the norm

l f x[°.4 du дхг

ЙХ[0,Г]

+ IMI

where N = {ж: x e Q , x x = 0 } , « > 1 , ||* Ilfx[0,2] tbe norm in ^s(i3 x [ 0 , T])»

and let Ж\{й x [0, T]) be a subspace with the norm

|ЙХ[0,Ж]

V?

|йх[0,2'] du dxx

йх[0,Г]

+ IM!Vxio.y]

s+1/2

Theorem. 1° I f

f e X Î M G x l 0 ,^ ]), <P

и0^{е Н}8( Г х [ 0 , T]), U x € Hs+ll2(dü X [0 , jF]) , ^{S > 1} and conditions (2)-(3) are fulfilled, then ue Ж\{й x [0, T]) and

(4) w e ' « ) < + n? ii" + ||«11]“ й°-л+ 11«.„п^х[о-^),

Ж8 ^Ж8^{- 1} Ж8 ^Ж8^-\

where G does not depend on u.

In particular, for the energy integral the following inequality holds ipxioj] = с ш \ & ’т' + ы а 2

1strq ^ met ii«iii“ x[0’T,+ .„Ilf*!”'1'!) 2° ^jFor every

[f, <p, у,, u0, X [0 , Т})®Ж\{П)®Ж\_х( ®)®Я.АГх [0 , T])®

®-ffs+1/2( a f i x [ o , T ] ) , s > 1, exists only one ue x [0, T ]) such that conditions (2)—(3) are ful­

filled.

Non-coercive mixed problem 191

P roof. By differentiation of the both sides of equation (2) with respect to xx we obtain

» ( u x x)it ~ l j U x 1 ~ f x x i u x^\dQx[0,T\ —

Uxx\t=0 — Фхх, (Uxx)t\t=0 — Wx1’

Hence, the function uXl = @ { f Xl, фХ1, y>Xl) is defined uniquely, Щ, <pXl, yXl) e H s{Q x [0 , T]) and

(6) I K J I? x[0' :rl < 0 ,(11 /*, i f - ? ' 11 + I K I l f + I f e j l f - ! + ll«ilfia:xtM1)-

Hence

(6) и f ^1? Фххчipx^)àx1 -\-w(x', t),

where w(x’, t) can be an arbitrary function of class HS(N x [0, T]), also belongs to JSs(ü x [0, T]).

We shall show that condition (3) defines the function w(xr, t) uniquely.

Since the function и defined by (6) should be a solution of problem (2)-(3), it should be

(7) wtt- L xw

where

■ ~^{fxX1 ^{^1} 1 ^Фхх1 ^{Фхх) •}

д d2

Formula L = L x H---h + a — r- was used here, where lx — first

дхх dx[

order differential operator with respect to x\ L x — the elliptic operator with respect to x', coefficients of L x and lx do not depend on x x, a = a(x').

It is easy to notice that dF/dxx = 0.

The function w ( x ' , t) has to be a solution of the following mixed problem wtt~ L xw = ( / + lx В -f a@Xl) 1^=0, w |t_0 = Ф k=o >

W t\t = 0 = ф\хх= 0 1 W \rx[0,T] — U 0 ’

Hence, the second part of the theorem follows..

It is easy to see that

(8)

+ l M f +

Ilf-, IK

< C »(ll/lf_1IM’1 +m°H4f'+ M f + M f - , + II«ollfxC0’al)

<

СЖг№

+ f e . l f - , » + W f - , + 1 К 1 1 Г (М1 + I K I É Î ? ' 11) •

192 H. Kolakowski

The estimate (4) follows from inequalities (б), (8) and from the inequality (see [1])

INI?X[0,T1 < \ \ u -w ÿ * № + \\u,\)ï*№.

< 0 (|| f UH |lOxt0-11 + |M|f* < CUlN^iir[оя + IMlf *м ) ■ 0

The author wishes to thank Professor B. Bojarski for suggesting this problem.

References

[1] Ю. В. Егоров, В. А. Кондратьев, О задаче с косой производной, Матем. сб.

78(120), № I (1969), стр. 148-176.

[2] Л. Хёрмандер, Линейные дифференциальные операторы с частными производными, Москва 1965.