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

Let Omega subset of R-n be a bounded starshaped domain. In this note we consider critical points (u) over bar is an element of (ξ) over bary + W-0(1,p) (Omega; R-m) of the functional F(u, Omega) := integral(Omega) f(delu(y))dy, where f : R-m x n --> R of class C-1 satisfies the natural growth \f (xi)\ less than or equal to c(1+ \xi\(p)) for some 1less than or equal top<&INFIN; and c>0, is suitably rank-one convex and in addition is strictly quasiconvex at (ξ) over bar is an element of R-m x n. We establish uniqueness results under the extra assumption that F is stationary at (u) over bar 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 Muller Sverak (2003).