Local Lipschitz continuity of solutions to a problem in the calculus of variations

摘要

This article studies the problem of minimizing f(Omega) F(Du) + G(x, u) over the functions u is an element of W-1,W-1 (Omega) that assume given boundary values phi on partial derivative Omega. The function F and the domain Omega are assumed convex. In considering the same problem with G = 0, and in the spirit of the classical Hilbert-Haar theory, Clarke has introduced a new type of hypothesis on the boundary function phi: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the classical bounded slope condition of Hartman, Nirenberg and Stampacchia, is satisfied if phi is the restriction to partial derivative Omega of a convex (or concave) function. We show that for a class of problems in which G(x, u) is locally Lipschitz (but not necessarily convex) in u, the lower bounded slope condition implies the local Lipschitz regularity of solutions. (C) 2007 Elsevier Inc. All rights reserved.