Analyse (Cours de mathematiques) (French Edition) by Jean Marie Arnaudies PDF

By Jean Marie Arnaudies

ISBN-10: 2040165010

ISBN-13: 9782040165017

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Extra info for Analyse (Cours de mathematiques) (French Edition)

Example text

We note that in the Banach space case, the derivative g' is a continuous mapping from B to its adjoint space B*. At each point x of Me> its tangent space Tx (Me) can be identified with the subspace of B given by Tx (Me){v 1 vE B, (g'(x), v) = O}, where we use the notation (w, v) for the pairing between w in the space B* and v in the space B. The injection of Tx (MJ into B which we have just described yields a canonical Finsler structure on the manifold Me where the norm of v in Tx (Me) is the corresponding norm in B of its canonical image in B.

Let G and K be tunctions satistying the A ssumptions given above, and let A and B be the corresponding Euler-Lagrange operators A(u) = B(u} = L (-1)1"'ID"'G",(x,~m(u)), L (-1)IßIDßK ß (x,tpm_l(U)). 1",I;;;;m IßI;;;;m-l I t V is a closed subspace ot Wm,p (Q) with the Sobolev I mbedding Theorem valid on Q, then u trom V is said to be an eigentunction tor A(u) =AB(u), Areal, with the natural boundary conditions corresponding to V, (or brietly, u is an eigentunction tor (A, B) with respect to V and with eigenvalue A), it the tollowing holds: a (u, v) = Ab (u, v), tor all v in V.

4). Sinceeach Me,n is a c10sed bounded set in a finite dimensional space, it is compact. Since the Me, n increase with n, it follows immediately that cat (X n; X) do not decrease with increasing n, and that for every n cat (X n; X) ;;;; comp cat (X) . It suffices therefore to show that cat (X n ; X) -J>comp cat (X). This follows directly from the result of part (a), however, since given a Nonlinear Eigenvalue Problems and Group Invariance 55 compact subset K of X with cat (K; X) ~ k for a given integer k, we may deform it over X into a compact subset K n of Xn- Hence cat (Kn;X) ~k, and the limit relation follows.

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Analyse (Cours de mathematiques) (French Edition) by Jean Marie Arnaudies


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