Journal article

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form Phi(u) = integral(Omega) L(x, u(x),del u(x)) dx on the Sobolev space H-0(1)(Omega) where Omega is a bounded open subset of R-N with N >= 3 and 0 is an element of Omega The hypotheses on L ensure that u 0 is a critical point of Phi, but allow the Lagrangian to be singular at x = 0. It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of Phi at u 0), Legendre (ellipticity at u 0) andWeierstrass [ strict convexity of L(x, s, xi) with respect to xi] from the calculus of variations are not sufficient ensure that u 0 is a local minimum of Phi. Using recent criteria for the existence of a potential well of a C-1- functional on a real Hilbert space, conditions implying that u 0 lies in a potential well of Phi are established. They are shown to be sharp in some cases.


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