Partial regularity for minimizers of variational integrals with discontinuous integrands (Q1814666)

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.