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Blowup, compactness and partial regularity in the calculus of variations. (English) Zbl 0626.49007
Let n, N denote positive integers, \(M^{n\times N}\) be the space of all real \(n\times N\) matrices, \(\Omega \subset {\mathbb{R}}^ n\) is open bounded and smooth and consider the functional \(I(v)=\int_{\Omega}F(Dv)dx\), where \(v: \Omega\to {\mathbb{R}}^ N\), \(Dv=[\partial v^ i/\partial x_{\alpha}]\), \(1\leq \alpha \leq n\), \(1\leq i\leq N\). The regularity of minimizers of I(.) among all appropriate functions subject to some given boundary conditions is considered. The main contribution is to point out that the so-called “Caccioppoli-inequality” is unnecessary; in fact, partial regularity can be proved by a blow-up argument combined with a careful analysis of the compactness of various sequences of functions.
Reviewer: S.Tersian

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
49J20 Existence theories for optimal control problems involving partial differential equations
26B25 Convexity of real functions of several variables, generalizations
35J65 Nonlinear boundary value problems for linear elliptic equations
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