By W.A. Coppel

ISBN-10: 0669190187

ISBN-13: 9780669190182

Balance and Asymptotic habit of Differential Equations (Heath Mathematical Monographs)

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**Additional info for Stability and asymptotic behaviour of differential equations**

**Example text**

A.. be the nth eigenvalue of A m G*, G • The idea of the proof is to extend any u* E if 1 (G*) by zero to au E if 1 (G), thus imposing constraints. Therefore A! (G), II vii= I B(v, v) = A1 • To get the same result for A~, we use Courant's mini-max principle. Let hiEYf'(G), 1 ~j ~ n-1, be fixed. Then or proving the theorem. 6 The analogue for the Neumann problem goes the other way round. Yt' 1 (G)we'split it up' into vEYf' 1 (G*) n Ye 1 (G\G*), thus 49 loosening constraints. Therefore, the nth eigenvalue in the collection of eigenvalues of both G* and G\G* is smaller than or equal to An.

This method does not assume 8 1 to be self-adjoint and in fact is often used to prove self-adjointness. Next let us consider the original Dirichlet problem (homogeneous equation, inhomogeneous boundary values), which led to the Dirichlet principle. tf 1 such that (i) (ii) u-gEX 1 VvEX 1 , (u,v) 1 =0. This problem, too, can easily be solved using the projection theorem. X 1 is a closed subspace of Jf 1 , so we get Jf 1 = it' 1 EB it' f. Let u be the projection of g in it' f. Then u solves the Dirichlet problem.

To clarify this we notice £C(A*)={ueJfl3hEJf, Vve£C(A), = {ueJf I AueJf} A*u:= h= -Au. (u,Av)=(h,v)} 24 Thus A c A c A*. If G = IR", using Fourier transformation we get 0'! 2 Jf" 2 (1R") = ff 2 (1R"), so that A= A*. But generally this is not true. Next we want to find self-adjoint extensions of A. Because of (Au, u) = (Vu, Vu) ~ 0, A is semi-bounded and the Friedrichs extension exists. To make the calculations easier, we first study B:=A+l yielding (Bu, v) = (u, vh, (Bu, u) = llulli ~ llull 2 .

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