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Partial regularity for minimizers of variational integrals with discontinuous integrands. (English) Zbl 0863.35022
The author proves the partial regularity for vector-valued minimizers \(u\) of the variational integral \[ \int_\Omega [f(x,u,Du)+g(x,u)]dx, \] where \(f\) is strictly quasiconvex, of polynomial growth and continuous and \(g\) is a bounded Carathéodory function. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(n\geq 2\), \(u:\Omega\to\mathbb{R}^N\), \(N\geq 1\) and \(Du(x)\in \mathbb{R}^{N\times n}\) denotes the gradient of \(u\) at the point \(x\in\Omega\). An elementary proof for the special case of strict convexity and quadratic growth of \(f(x,u,\cdot)\) is presented.

35B65 Smoothness and regularity of solutions to PDEs
49N60 Regularity of solutions in optimal control
35J50 Variational methods for elliptic systems
Full Text: DOI Numdam EuDML
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