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

This article studies the problem of minimizing ∫ΩF(Du)+G(x,u) over the functions u W1,1(Ω) that assume given boundary values on ∂Ω. The function F and the domain Ω 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 : 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 is the restriction to ∂Ω 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.