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 ϕ 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 ϕ: 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 ϕ 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.