Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case

Abstract

Under general growth assumptions, that include some cases of linear growth, we prove existence of Lipschitzian solutions to the problem of minimizing ${\int }_a^{b}L(x(s),x^{\prime }(s))ds$ with the boundary conditions $x(a)=A$, $x(b)=B$.

Résumé

Dans l’article on démontre l’existence de solutions Lipschitziennes du problème de minimiser ${\int }_a^{b}L(x(s),x^{\prime }(s))ds$, $x(a)=A$, $x(b)=B$, avec des conditions faibles, de croissance, qui comprennent des cas de croissance linéaire.