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Properties of Bilinear Forms on Hilbert Spaces Related to Stability Properties of Certain Partial Differential Operators

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Properties of Bilinear Forms on Hilbert Spaces Related to Stability Properties of Certain Partial Differential Operators JOURNAL OF hlATHEhlATICAL ANALYSIS AND APPLICATIONS 20, 124-144 (1967) Properties of Bilinear Forms on Hilbert Spaces Related to Stability Properties of Certain Partial Differential Operators N. SAUER National Research Institute for Mathematical Sciences, Pretoria, South Africa Submitted by P. Lax INTRODUCTION The continuous dependence of solutions of positive definite, self-adjoint partial differential equations on the domain was investigated by Babugka [I], [2]. In a recent paper, [3], Babugka and Vjibornjr also studied the continuous dependence of the eigenvalues of strongly-elliptic, self-adjoint partial dif- ferential operators on the domain. It is the aim of this paper to study formalisms in abstract Hilbert space by means of which results on various types of continuous dependences can be obtained. In the case of the positive definite operator it is found that the condition of self-adjointness can be dropped. The properties of operators of this type are studied in part II. In part III the behavior of the eigenvalues of a self- adjoint elliptic operator under various “small changes” is studied. In part IV some applications to partial differential operators and variational theory are sketched. I. BASIC: CONCEPTS AND RESULTS Let H be a complex Hilbert space. We use the symbols X, y, z,... for vectors in H and the symbols a, b, c,... for scalars. The inner product in H is denoted by (~,y) and the norm by I( .v 11 = (x, x)llz. Weak and strong convergence of a sequence (x~} to an element x0 E H are denoted by x’, - x0 and x’, + x,, , respectively. A functional f on H is called: (i) continuous if it is continuous in the strong topology on H, (ii) weakly continuous (w.-continuous) if it is continuous in the weak topology on H. 124 PROPERTIES OF BILINEAR FORMS 125 If f is real-valued, it is said to be weakly lower semicontinuous (w.l.s.- continuous) if lirnirff (xn) > f (x) whenever .v, - x. 1. Bilinear forms on H x H 1. DEFINITION. Let Q be a bilinear form on H x H. Q is bounded ;f for some M > 0 I Q&Y) I d JJI II * II * II Y II for all x,y E H. (1) Q is positive dejinite if for some m > 0 ReQ(x, x) > m 11x II2 for all x E H, (2) and dejnite if for some m > 0 I 0(x, 4 I 2 m II x l12. (3) We shall usually refer to a bilinear form as a “form.” A Hermitean form Q (i.e., Q(x, y) = Q(y, x)) is: (i) positive on H if Q(x, x) > 0 for all nonzero x E H, (ii) nonnegative on H if Q(x, x) 3 0 for all s E H. Concepts such as negative, nonpositive, etc., are defined analogously. If M is a linear manifold (i.e., linear subspace) in H, and if Q is a bounded Hermitean form on H then we define the subspaces (i.e., closed linear sub- spaces) MQ = {x : .v E H and Q(x, y) = 0 for ally E M}, 2. THEOREM ([4], [5]). Let Q be a bounded Hermitean form on H. Then three mutually orthogonal and Q-orthogonal subspaces Hz , Hf and H,” of H exist such that H = H: @ H,” @ H,” and such that Q is positive on H,” and negative on H!. 3. DEFINITION. Let Q be a bounded Hermitean form on H. We say that Q is elliptic on H if (i) Q(x, x) is w.l.s.-continuous and . (ii) x, - x and Q(xn , x,) -+ Q(x, x) imply x,, -+ x. 126 SAUER If Q is bounded (but not necessarily Hermitean) we say that Q is elliptic on H if Q(O) defined by Q’O’(x,Y) = i [Q&Y) + !x?(r, 41 (4) is elliptic on H. 4. LEMMA ([4]). If Q is Hermitean and elliptic on H then the space H! @ HoQ is finite-dimensional. We also have the following characterization of ellipticity. 5. THEOREM ([4]). A Hermitean form Q is elliptic on H sf and only sf Q is the sum of two Hermitean forms P and K with Ppositive definite and K(x, x) w.-continuous. 6. COROLLARY. Let {x~ : a: E A) be a net (i.e. generalized sequence) of elements of H and let the Hermitean form Q be elliptic on H. If xti - x E H and if the limit lim,Q(x, , XJ exists, then lim,Q(x, , x,) > Q(x, x). If lim, Q(G , xJ = Q(x, x), then xE -+ x. PROOF. We need only prove the inequality for a positive definite form Q. It follows from the identity: 86~ , x,) - Q(x, x) = Q(x& - x, xm - 4 + Q(x, x, - x) i Q&x - x, x) Q.E.D. For definite. bounded forms we obtain 7. THEOREM. If Q is bounded and definite, then (4 x, - x and Q(xn , x,) + Q(x, x) imply x,, + x. (b) Q(xn , x,) + 0 implies x, + 0. (c) For any x* E H* there is a unique x E H such that Q(z, x) = x*z for all z E H. (c) is known as the Lax-Nilgram lemma [6]. 8. DEFINITION. If Q is Hermitean we denote the quadratic form Q(x, x) by Q(x)- II. STABILITY PROPERTIES OF DEFINITE FORMS We suppose that Q is a bounded definite form on H x H with H a complex Hilbert space. We shall refer to problem P(x*, Q, H) as the problem of finding x E H such that Q(z, x) = x*z for all ZEH;X*EH*. PROPERTIES OF BILINEAR FORMS 127 1. Inner and Outer Approximation-s of a Space 1. DEFINITION. Let X be a family of proper subspaces of H. We say that X is an inner approximation of H if (a) H = sp{K : K E z@> (s$ = smallest subspace 1 all K E .#) and (b) the partially ordered set (2, C) is directed. 2. THEOREM. Let x* E H* be given and let & be an inner approximation of H. Dejke the net {x(K) : k’ E ti} of e1 ements of H by letting x(K) be the solution of P(x*, Q, K). Then, if x E H is the solution of P(x*, Q, H), 19 11s - x(K) 11= 0. (5) PROOF. We see that the net (x(K) : K E .%} is bounded for m II x(K) 11’< I Q(x(K), x(K)) I = I x*x(K) I < II x* II * II x(K) II. Hence, by the reflexiveness of H there is a weakly convergent subnet {x(K,) : /E ?Z} where (9, <) is directed. Let x(KG) - x E H. We see that CC solves the problem P(x*, Q, H) for if z E sp{K : K E Z} (sp = smallest linear subspace > all K), then, since (Z, C) is directed, there is K1 E 2 such that z E K1 . Since (9, <) is directed it follows from the definition of subnet and weak convergence that, for every E > 0, there is 8, E 9 such that / > /s implies that K/1 K1 and ( Q(z, x(Kt)) - Q(z, x) ( < E. But Ktl K, implies that Q(.z, x(K,)) = x*z; i.e., Q(z, x) = x*z for all z~sp{K:K~~}.Butsp{K:K~~}isdenseins~{K:K~~}=H and hence Q(z, X) = x*z for all z E H. But this argument is valid for every weakly convergent subnet of {x(K) : K E X’}, i.e. every weakly convergent subnet converges weakly to X. It follows by a simple argument that {x(K) : K E Z} converges weakly to x. Finally, since Q(x, X) - Q(x(K), x(K)) = x*(x - x(K)), we see that Ii$iQ(x - x(K), x - x(K)) = 0 and hence (5) holds. Q.E.D. An argument analogous to the proof of Theorem 2 can be used to prove the validity of the Lax-Milgram lemma. If it can be shown that the problem P(x*, Q, K) is solvable for every K E Z, then it follows that the limit x of the net {x(K)} is a solution of P(x*, Q, H). This has indeed been done by the writer, [7], for the case where H is a reflexive Banach space by taking 2 as the family of all finite dimensional subspaces of H. If H is a separable Hilbert space with Schauder basis {ui , us ,...}, the family 2 can be chosen to 128 SAUER consist of the finite dimensional subspaces Hl C Hz C ... where H, = sp{u, ,..., nn}. In this case a constructive proof of the Lax-Milgram lemma is obtained (see e.g. [8] and [9]). 3. DEFINITION. Let L be a proper subspace of H and let ,8 be a family of subspaces of H such that L is a proper subspace of every member of .@. .%! is called an outer approximation of L if (a) L = n (K : K E Z} and (b) the partially ordered set (Y, I) is directed. 4. THEOREM. Let x* E H” be given and let Y be an outer approximation of L C H. DeJilze the net {.x(K) : K E X} of e1 ements of H by letting x(K) be the solution of P(x*, 0, K). Then, if s EL is the solution of P(x*, Q, L), ‘$I (1x - N(K) // = 0. (6) PROOF.
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