Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Abstract

Let ω ⊂ Rn be a bounded starshaped domain. In this note we consider critical points ū ∈ χ̄y + W 01,p(ω;Rm) of the functional ℱ(u,ω):= ∫ω f(∇u(y))dy, where f : R m×n → R of class C1 satisfies the natural growth |f(χ)| ≤ c(1 + |χ|p) for some 1 ≤ p < oo and c > 0, is suitably rank-one convex and in addition is strictly quasiconvex at χ̄ ∈ Rmxn. We establish uniqueness results under the extra assumption that ℱ is stationary at ū with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Šverák (2003).