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MiS Preprint
59/2001
Partial regularity of strong local minimizers in the multi-dimensional calculus of variations
Jan Kristensen and Ali Taheri
Abstract
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $F:\mathbb{R}^{N\times n} \to \mathbb{R}$ a given strongly quasiconvex integrand of class $C^2$ satisfying the growth condition $$|F(\xi)|\leq c(1=|\xi|^P)$$ for some c>0 and $2\leq p < \infty $. Consider the multiple integral $$I[u]=\int_\Omega F(\nabla u)$$ where $u \in W^{1,p} (\Omega , \mathbb{R}^N)$. The main result of the paper is that any strong local minimizer of $I[\cdot]$ is of class $C^{1,\alpha}$ for any $\alpha \in (0,1)$ on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular. This is shown by a class of examples that are obtained by suitably modifying the arguments of S. Müller and V. Sverak.
