Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations

Abstract

SynopsisWe demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.