Champs Hamiltoniens, relaxation et existence de solutions en calcul des variations. (Hamiltonian fields, relaxation and existence of solutions in the calculus of variations)
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.
scientific article; zbMATH DE number 4048499
scientific article