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**Example text**

The variational formulation of the Dirichlet problem is: Given / G (ff^*(n)V, 5i G H3/2(dn), and g2 G Hl/2{dQ), find u G F 2 -*(Q), such that u\aa = Si, f^lan = 92, and a(u, v) =< / , v >, Vv G ffo'*(n). (L53) where (iJ 0 '*(Q) 1 is the dual space of H0'*(Q,). Theorem 23. 53) admits a unique solution. Proof. By the inverse trace theorem there exists w0 G H2'*{Q) satisfying the boundary condition, so u — UQ satisfies homogeneous boundary conditions. Applying the Green's formula we get fat*. A , * - f j {( g ) 2 + 2( £ ^ j + ( g ) 2 } * _ CMi in iJ 0 '*(fi).

90) follows. • By the inverse trace theorem of Sobolev spaces there exists u\ € (H1^)) with a compact support on Q, such that ||ui||i < \\g — Woo||i> and u\ = g — u^ on dQ. Let R be large enough such that B(0,R) D supp Mi, and let fix = Q n B(0, R). We apply Lemma 20 to get a function u2 G ( H ^ f t i ) ) 3 such that V • u2 = - V • ui and ||w2||i < C||V • iti||o- The Lemma 20 is about an exterior domain, however it can be verified that it is 48 Numerical Methods for Exterior Problems also applied to bounded domains.

The residual spectrum is ar(A) = {A G a{A); R(A - XI) ^ H,{A- XI)one - to - one}. Theorem 35. Let A be a self-adjoint operator. Then X G o~p{A) if and only if E(X) — E(X — 0) 7^ 0, crr(A) is a null set, and the necessary and sufficient condition for X G cr(A) is that Ve > 0, E(I*) ^ 0, where J» = (A — £, A + e). Let us now investigate the initial boundary value problem of the heat equation. We remark that although the above definitions and results are all for complex numbers, we will assume the functions are real in the remaining part of this section, and it is easy to see that the following result is valid for both real and complex cases.