Publication

Criteria for the existence of a potential well

Charles Stuart
2017
Journal Articles

Abstract

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.

Official source

https://infoscience.epfl.ch/entities/publication/8869778a-44e4-4f26-a0ec-358c5d99428a

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Charles Stuart
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/8869778a-44e4-4f26-a0ec-358c5d99428a

About this result

Related concepts (32)

Fréchet derivative

In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces.

Gateaux derivative

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

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Criteria for the existence of a potential well

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