Strict convexity and the regularity of solutions to variational problems^∗

Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy
arrigo.cellina@unimib.it

Received: 12 December 2014

Abstract

We consider the problem of minimizing where Ω is a bounded open subset of ℝ^N and L is a convex function that grows quadratically outside the unit ball, while, when | ∇v | < 1, it behaves like | ∇v | ^p with 1 <p< 2. We show that, for each ω ⊂ ⊂ Ω, there exists a constant H, depending on ω but not on p, such that both in particular, for every i = 1,...N, we have .