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

Abstract:

Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain. In this note we consider critical points $\bar {u} \in \bar {\xi } y + W_0 ^{1,p} (\Omega ; {\mathbb R}^m)$ of the functional \[ {\mathcal F}(u, \Omega ) := \int _{\Omega } f( \nabla u(y)) dy, \] where $f: {\mathbb R}^{m \times n} \to {\mathbb R}$ of class $\mathrm {C}^1$ satisfies the natural growth \[ |f (\xi )| \le c (1 + | \xi |^p) \] for some $1 \le p < \infty$ and $c>0$, is suitably rank-one convex and in addition is strictly quasiconvex at $\bar {\xi } \in {\mathbb R}^{m \times n}$. We establish uniqueness results under the extra assumption that ${\mathcal F}$ is stationary at $\bar {u}$ 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).