Fréchet space

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces. A Fréchet space X is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning that every Cauchy sequence in X converges to some point in X (see footnote for more details). :Important note: Not all authors require that a Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely, if the topology of a locally convex space X is induced b

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Bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem

Charles Stuart

We consider bifurcation from the line of trivial solutions for a nonlinear eigenvalue problem on a bounded open subset, Omega, of R-N with N >= 3, containing 0. The leading term is a degenerate elliptic operator of the form L(u) = del . A del u where A is an element of C((Omega) over bar) with A > 0 on (Omega) over bar {0} and lim(x -> 0) A(x)/vertical bar x vertical bar(2) is an element of (0, infinity). Solutions should satisfy u = 0 on partial derivative Omega and the energy associated with L should be finite: integral(Omega) A vertical bar del u vertical bar(2)dx < infinity. The nonlinear terms are of lower order, depending only on u and del u. Under our hypotheses the associated Nemytskii operators are not Frechet differentiable at the trivial solution u = 0. (C) 2014 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2015

Criteria for the existence of a potential well

Charles Stuart

We consider a critical point u(0) of a functional f is an element of C-1 (H, R), where H is a real Hilbert space, and formulate criteria ensuring that u(0) lies in a potential well of f without supposing that f' is Frechet differentiable at u(0). The derivative is required to be Gateaux differentiable at u(0), but positive definiteness of f ''(u(0)) does not even ensure that f has a local minimum at u(0) when f' is not Frechet differentiable at u(0). This issue is also discussed in the context of the energy functional for a parameter dependent nonlinear eigenvalue problem and then for a particular case involving a degenerate elliptic Dirichlet problem on a bounded domain in R-N. (C) 2017 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2017

In an open, bounded subset Omega of R-N such that 0 is an element of Omega we consider the nonlinear eigenvalue problem -Sigma(N)(i,j,=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + V(x)u + n(x,del u)+ g(x, u) = lambda u in Omega integral(Omega) u(2) + Sigma(N)(i,j=1) A(ij)partial derivative(j)u partial derivative(i)u dx < infinity and u = 0 on partial derivative Omega, where V is an element of L-infinity(Omega) and the nonlinear terms n and g are of higher order near 0 so that the formal linearization about the trivial solution u 0 is - Sigma(N)(i,j=1) partial derivative(i){A(ij)(x)partial derivative(j)u} + Vu = lambda u. The leading term is degenerate elliptic on Omega because it is assumed that there are constants C-2 >= C-1 > 0 such that C1 vertical bar x vertical bar(2)vertical bar xi vertical bar(2)

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