Champs Hamiltoniens, relaxation et existence de solutions en calcul des variations. (Hamiltonian fields, relaxation and existence of solutions in the calculus of variations)
type: Item
instance of: scholarly article
title: Champs Hamiltoniens, relaxation et existence de solutions en calcul des variations. (Hamiltonian fields, relaxation and existence of solutions in the calculus of variations)
zbMATH Open document ID: 0643.49022
DOI: 10.1016/0022-0396(87)90164-1
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Let us consider the following problem of the calculus of variations in \(R^ 2\) \[ {\mathcal P}(T,\alpha,\beta): Inf\{\int^{T}_{0}(g(\dot x(t))+h(x(t)))dt;\quad x\in (W^{1,q}(0,T))^ 2,\quad x(0)=\alpha,\quad x(T)=\beta \}, \] where \(1<q<+\infty\) and g and h are nonconvex smooth functions. Let \(g^{**}\) be the greatest convex function satisfying \(g^{**}\leq g\) and let us consider the relaxed problem \({\mathcal P}{\mathcal R}(T,\alpha,\beta)\) associated with \({\mathcal P}(T,\alpha,\beta)\), i.e. the problem obtained from \({\mathcal P}(T,\alpha,\beta)\) by replacing g with \(g^{**}\). By using the maximum principle for \({\mathcal P}{\mathcal R}\), the author associates with any solution x of \({\mathcal P}{\mathcal R}\) a trajectory in the phase space, and he studies the trajectories of \({\mathcal P}{\mathcal R}\) in the case where \({\mathcal P}\) does not admit a solution. He proves the following interesting property: there exists at least one solution \(\bar x\) of \({\mathcal P}{\mathcal R}\) such that if the restriction of g to some line \(\Delta\) through \(\bar x\dot{\;}(t)\) does not satisfy the classical Weierstrass condition, then the restriction of h to the line direction \(\Delta\) going through \(\bar x(t)\) is strictly convex. By using this result, the author proves a necessary and sufficient condition on (g,h) in order that P has at least one solution for every boundary conditions.
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49N15
zbMATH DE Number: 4048499
zbMATH Keywords: relaxed problem
zbMATH Keywords: maximum principle
author: J-P Raymond
reviewed by: Ioan I. Vrabie
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90164-1
OpenAlex ID: W2094887889
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description: scientific article; zbMATH DE number 4048499