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MiS Preprint
29/2001
Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations
Ali Taheri
Abstract
Let $\Omega \subset \mathbb{R}^n$ be a starshaped domain. In this note we consider critical points $\overline{u} \in \xi x = W^{1,p}_0 (\Omega ; \mathbb{R}^m)$ of the functional $$F(u,\Omega) := \int_\Omega f (\nabla u(x)) dx$$ where $f:\mathbb{R}^{m\times n } \to \mathbb{R}$ of class $C^1$, is suitably rank-one convex, and in addition strictly quasiconvex at $\xi \in R^{m\times n}$. We establish uniqueness results under the extra assumption that $F$ is stationary at $\overline{u}$ with respect to variations of the domain. These results should be compared to the ones by Knops and Stuart and recent counterexamples produced by Müller and Sverak.
