Proc. Indian Acad. Sci. (Math. Sci.), Vol. 98, No. 1, June 1988, pp.63-69. �9 Printed in India.

Nonlinear random elliptic boundary value problem

MOHAN JOSHI Department of Mathematics, Indian Institute of Technology, Bombay 400 076, India

MS received 23 January t987

Abstract. In this paper we obtain a solvability result for random elliptic boundary value problem. Our result extends Browder's [2] main theorem to the random case. We use author's [4] fixed point theorem for this purpose.

Keywords. Elliptic boundary value; fixed point theorem.

1. Introduction

In this paper we extend Browder's [2] main result to obtain solvability of random variational boundary value problem involving system of the form

A(w)x= ~ (-1)l~lD%4~(w,t,x,...,Dmx), (*) I~[~m

where the coefficients A~ are random having at most polynomial growth. By using author's [4] random fixed point theorem, we first obtain an abstract result

for a class of random operator equation of the type

G(w; x(w), x(w) ) = z.

Here G:fl x Y x X-+X* is a random operator, ~1 a probability measure space, X, Y the appropriate Banach spaces and z~X* a given element. This result is then applied to discuss the solvability of boundary value problem involving system of the form (*).

The abstract result is of interest by itself. A similar result has been obtained by Itoh [3] but with relatively strong conditions on the operator G. We therefore discuss it separately in w 3. In w 4 we consider the random elliptic boundary value problem which is followed by a concrete example. Section 2 gives preliminary definitions concerning monotone operators and random solutions.

2. Preliminaries

Throughout this paper (f~,fl,/~) denotes a probability measure space. For xeX and yeX*, (y, x) will denote the duality pairing between X and X*. If X is Hilbert space ( , ) indicates the inner product in X. In all our considerations Banach spaces are assumed to be real and separable.

A mapping g:f~ ~ X is said to be measurable, if for any closed subset C of X, g-1(C) is in ft. Any X-valued measurable function is called a random variable. If Y is a

63

64 Mohan Joshi

Banach space and D a subset of X, T:f~ • D ~ Y is called a random opera tor if for any x e D , T(.) x is measurable. An equat ion of the type T(.)x( ' )= y(-), where y, a random variable, is called a r andom opera tor equation. An X-valued random variable x(w) which satisfies/L{w: T(w)x(w) = y(w)} = 1, is said to be random solution of the above equation.

An opera tor T:X--* Y is said to be bounded if it maps bounded sets into bounded sets. T is demicont inuous if x , ~ x implies that T x , converges weakly to Tx. T is said to be compact if it maps bounded sets into precompact sets. T: X ~ X * is called mono tone if ( T x -- Ty, x - - y ) >>. 0 for all x, y e X . With these definitions we have the following theorems.

Theorem 2.1 [2]. Let G be a demicontinuous mapping of X into X* and let there exists a continuous function c(r) with c(r) > 0 for r > 0 and lira c(r) = oo such that

r ~ o 9

( Gu - Gv, u - v ) >~ c( ll u - v ll ) ll u - v ll for all u, v in X. Then

(a) G maps X onto X*. (b) G is I-I , G -1 is defined on all o f X * and is bounded. (c) G -1 is continuous from X* to X and there exists a real valued monotone non-decreasin9 function h(r) such that

j[ G - i y _ G- 1Yl l[ ~< h( il Y - Yx II ) for y, Yl in X*.

In the following B, is the closed ball {x: II x II ~< r}, OBr its boundary with interior/~r.

Theorem 2.2 [4]. Let (f2, fl, p) be a probability measure space and X a separable reflexive Banach space, T:I) x B, ~ X a compact and continuous random operator. Suppose for a f ixed w e l l T(w) satisfies the condition: (<~) l f for some xec~B r the operator equation T(w)x = ctx holds, then r .~ 1. Then there exists a random solution x(w) in B, o f the operator equation T(w)x(w) = x(w).

3. Abstract theorem

Theorem 3.1. Let X be a separable reflexive Banach space with dual X*. Let Y be a second Banach space with X c Y and I[ x I[ r <~ [I x II x for all x ~ X . Let G :f~ x Y x X ~ X* be a random mapping satisfying the following conditions:

(a) For every w e ~ , there exists a continuous mapping c(w;r) with lim c(w,r)= oo and c(w; r) > 0 for r > 0 and a constant R > 0 such that "~ ~

( G(w;y, x l ) - G(w; y, x2) ,x 1 - x 2 ) >~c(w; I lx l - x2 I I ) l lx l - x2 IIx

almost everywhere for all x 1, x 2 e X and y e Y with II y II y ~< R. (b) ( G(w; x, kx) - z, x ~ > 0 for all x e X with II x II x = R and every k >>. 1 ( z e X * a f ixed element). (c) For each we f t , x --, G(w; y, x) is demicontinuous for f ixed y e Y and y - , G(w; y, x) is compact and continuous for f ixed x e X .

Then there exists a random solution x ( w ) e X of the equation G(w, x (w)x(w))= z.

Nonlinear random elliptic boundary value problem 65

Proof. Let y ~ X be fixed such that II Y II x ~< R. Then y~ Y with II Y [I r ~< R and in view of condition (a) of the theorem it tbllows that the random mapping G(W; y,') satisfies the hypothesis of Theorem 1.1 for a fixed weft. Hence it implies that G(w;y, ') is 1-1 with G-X(w; y, ) defined and continuous on all of X*. By Nashed-Salehi theorem I-5] it now follows that G-l(w; y, .) is also random, i.e. given z EX* there exists a unique random solution x(w) of the equation G(w; y , x ( w ) ) = z . Denoting T(w)y we get a mapping T ( w ) : B R ~ X such that G(w;y, T ( w ) y ) = z . T(w) is random as T(w)y is a random variable for every y ~ B R.

We. now claim that for every wef t , T(w) is a compact and continuous mapping satisfying the condition (~<) of Theorem 1.2. We first observe that T(w) is bounded. This can be seen from the following inequality and the fact that y---r G(w;y ,x) is compact and hence bounded for every xeX~

c(w; !t T(w)y ]l)II T(w)y H <~ ( G(w; y, r (w)y) - G(w; y, 0), T(w)y - 0 )

= ( z - G(w; y, 0), T(w)y )

~< II z - a(w; y, 0)tl II T(w)y II,

which gives

c(w; II T(w) y II) ~ II z - G(w; y, O)If.

We now proceed to show that G(w) is compact. Let II y, 1I <~ R be any sequence. Since T(w) is bounded and X is reflexive, T(w)y, has a subsequence T(w)y,k converging weakly to x e X . We will show that this convergence is strong.

Since y ~ G ( w ; y , x ) is compact for every x E X , it follows that G(w;y,~,x) has a subsequence, which we will not differentiate, converging to zoeX* . Consider

c(w; II T(w)Y.k -- x II)11 T(w)y,k - x [1

<<, ( G(w; Y.k, T(w)y.~) - G(w; Y.k, x), T(w)y.k -- x )

= ( z -- z o + (z o -- G(w; y.k, x)), T(w)y,~ - x )

<~ ( z - z o, T(w)y,~ - x ) + II Zo - a(w; y.~, x)II II T(w)y.~ - x II.

Since T ( w ) y , ~ - x ~ O and z o - G ( w ; y , ~ , x ) ~ O , the right hand side of the above inequality goes to zero as n ~ ~ and hence II T(w)Y,k -- x II --' 0.

One can similarly prove that the mapping T(w) is continuous. To prove that T(w) satisfies condition ( ~< ), let T(w)y = c~y for some y in (')BR. Then G(w; y, T(w)y) = G(w; y, ey). But G(w; y, T(w)y) = z, so we have ( G(w; y, ey), y ) - (z, y ) = 0. That is, (G(w; y, ey) - z, y ) = 0 for yeOBR. In view of the assumption (b) on G, it implies that ~ < 1 .

Thus the random operator T(w) satisfies all the conditions of Theorem 1.2 and hence there exists a random solution x(w) of the fixed point equation T(w)x(w) = x(w). Using the definition of T(w), it follows that x(w) is a random solution of G(w; x(w), x(w)) = z.

Remark. In [3] Itoh has proved a similar result (Theorem 3.4). He has monotonicity condition on the map x ~ G(w; y, x) for each fixed y e Y. Our condition (a) is similar to this but it needs to be satisfied only for y e Y with II y It ~< R. Further, he assumed

66 Mohan 3oshi

complete continuity on y---,G(w;y,x) for each fixed x~X. But we impose only compactness and continuity on this map. We feel that this is an improvement.

4. Random elliptic boundary value problems

As an immediate application of the above result we now discuss the solvability of elliptic boundary value problem for system of the form

A(w)x= ~ (-1)l~lD'A,(w,t,x ..... D"x). (1)

Following Browder [1] we make use of the following terminology. The points of the open set Z c R" will be denoted by t = (t~, t2, . . . , t,) and the elementary differential operator

n ~ ~j

for an ordered n-tuple ~ = (~ , ~2,. . . , ~,) of non-negative integers with the order of the operator D ~ being written as

1~[= ~ %'R sm j = l

will denote the vector space whose elements are ~ = ~:[ctl ~< m and for any ~ER ~" and for

We define

l ~ < p < ~ , I~IP= ~ I~=1 p. lal~<m

w"'P(Z) = {X:xELP(Z),D=xELP(Z), ]0tl ~< m}.

W ' 'p is a reflexive separable Banach space with respect to the norm

"~ l l p

x = ~ , LI ,l,,p (l~l~mf I D~x[pdt)

(x, y) will denote the natural pairing between xeLP(Z) and yeLP(Z) with q = p(p - 1)- 1 We assume the following growth condition on each A~.

Assumption I. [A~(w; t, ~)[ ~< c~(w)l~l p - 1 + g=(w; t) almost surely, c~(w) > O, #,(w)EL q. Corresponding to (!) we may define the nonlinear Dirichlet form 0t(w; x, y) for each

pairing x, ye wm'P(Z) by

~(w;x,y)= ~ (A~(w;t,x .... ,O"x),O~y). Ictl <~ m

Because of Assumption 1, it follows that for a fixed weft, ct(w; x, y) is well defined for all x, yE W"P(Z) and satisfies an inequality of the type

10c(w; x, y)L ~< g(w; II x II re,p)II y II re.p,

Nonlinear random elliptic boundary value problem 67

where 9(w, r) is the function of the real variable r for a fixed weft. Since A~(w) is random it follows that e(w; x, y) is a random Dirichlet form.

In the following we assume that there exists a close subspace V of W"'P(Z) with C~(E) = V.

DEFINITION 4.1

Let f be an element of V*. Then x(w) is said to be the random solution of the variational boundary value problem for A ( w ) x - - f satisfying the null boundary conditions corresponding to the space V if

(a) x(w)eV and x(w) is random and (b) ctr ,; x(w), y) -- (f, y) for all ye V, almost everywhere.

To separate the dependence of Dirichlet form on the lower order from its dependance on the higher order we define a new Dirichlet form ~t(w; x, y, z) as

ct(w;x,y,z)= ~ (A~(w;t,x . . . . . D ~ - l x , Dmy),D~z). [ct[~<m

Again it follows from Assumption 1 that ct(w; x, y, z) is well defined for all x e W r"- LP(E) and all y,z in W"~'P(Z). By the Sobolev imbedding theorem, W m'p is contained in W"-Lp and the imbedding is compact and

I~(w; x, y, z) l ~< g(w; II x I1,.- 1.~)II y I1,.,~ 11 z I1,.,~-

Theorem 4.1. Let A(w) be a system of random differential operator of the form (1) satisfying Assumption I for a given value of p > 1. Let V be a closed subspace of Wm'V(~) such that C ~ ( E ) c V . Let the Dirichlet form ot(w;x,y,z) be as defined before for x6 W r"- I'P(E) and y, z in Wm'P(~,). Suppose all the following conditions are satisfied:

(a) For every weft, there exists a continuous mapping c(w; r) with lira c(w, r)= oo and r--~ ct3

c(w; r) > 0 for r > 0 and a constant R > 0 such that

~(w, x , y , y - z ) - ~ ( w , x , z , y - z ) >>. c ( w ; II y - z II re,p)II y - z hl,.,p

almost everywhere for all y, z in W m'p and II x II,-1,p ~< R. (b) ct(w; x, kx - f , x) > 0 almost everywhere for all x in V with II x II m,v = R and k >1 1.

Then for every f e V*, there exists a random solution x(w) in V of the variational boundary value problem for A(w)x = f.

Proof. Let X be the closed subspace V of W "'p with the Banach space structure endowed by W "'p. Let Y be the closure of V in W'-1 ,p with the norm of W m- I'P. Now consider the form ct(w; x, y, z) defined as before. Then we have

I~(w;x,y,z)l ~ g ( w ; Ilxlly)llyllxllzllx.

For a fixed x, y, ~(w; x, y, z) is a continuous random functional on X. Hence by Riesz representation theorem it follows that there exists a unique element G(w; x, y ) eX such that ct(w; x, y, z) = <G(w;x~y),z> for all wef~ and zeX .

Since ct(w; x, y, z) is random, it follows that G is random. In order to apply Theorem

68 Mohan Joshi

3.1, we need only verify conditions (a), (b) and (c). For a fixed weD, the verification of these conditions is similar to that of Browder [2] for the deterministic analogue and hence we omit the.details.

So it follows that there exists a random solution x(w) of the equation G(w;x(w), x(w)) = f. This in turn implies that ~(w; x(w), x(w), z)= (f, z) for all ze V. This proves the theorem.

Example. As a concrete example we consider the following elliptic boundary value problem:

b~ (w; t) + b2 (w; t) = Ao(w; t, x(t)) (2)

x( t )=0, te0Z; t=(tl, t2) , Y.=(0,1) x(0,1).

In order to apply Theorem 3.1, we take p = 2 , V= WoL2(•) and assume that the functions b x(w; t),b2(w; t),Ao(w; t, ~) are random functions and A o satisfies the Caratheodory conditions and a growth condition of the type

IAo(w;t,~)l ~<b(w)l~[ +9(w;t), b(w)>0 and g(w;t)eL2(~,).

Further, b 1 (w; t), b2(w; t)e L 2 (Z) and are such that b t (w; t), b 2 (w; t) >~ r > 0 almost surely. In terms of our earlier notation, the Dirichlet form ~(w; x, y, z) is given by

o~(w;x,y,z)=(Ao(w;t,x(t)),z(t))+(bl(w;, t)~,~)+~b2(w;t)~2,~2).Oz ~ [ t3y Oz xx

It is easily seen that

~(w; x, y, y - z) - ~(w; x, z, y - z)

Ot2

f~[(Oy Oz~2+(Oy Oz~2]d t

2n 2 '~ ~>r l~_~-2)l[y--zll~.2 foral l xeL2(y~)

(by Poincare inequality).

Also assume that there exists an R > 0 such that A 0 satisfies the following inequality

f~ A~ ~ -~t f~ for Ilx[ll.2 = R,

(~t is some positive constant).

Then we have

2 I/ 2kr~t2 "~ ~(w;x, kx, x)>~ --~tI[XlrL2 + {1\ ~-~n 2 ) [IxlI2 2

Nonlinear random elliptic boundary value problem 69

provided

2rTt2 "~k ~> - : ( + l + 2 n f ) ilx112"2

> 0 for i ixPit ,2=R, k > l ,

2rTz 2

Since all the conditions of Theorem 4.1 are satisfied, it follows that there exists a r andom solution x(w) of the variational boundary value problem corresponding to equation (2).

Acknowledgement

This research was partially supported by CSIR, New Delhi.

References

[1] Browder F E, Probleraes nonlineaires (Montreal: Universite Presse) (1966) [2] Browder F E, Nonlinear elliptic boundary value problems I1, Trans. Am. Math. Soc. 117 (1965) 530-550 [3] ltoh S, Nonlinear random equations with monotone operators on Benach spaces, Math. Ann. 236

(1978) 133-I46 [4] Joshi M, Nonlinear random equations with P-compact operators in Benach spaces, Indian J. Pure

AppL Math. II 0980) 791-799 [5] Noshed M S and Salehi H, Measurability of generalized inverses of random linear operators, SIAM

J. Appl. Math. 25 (1973) 681-692