The singular set of Lipschitzian minima of multiple integrals
This paper deals with the partial regularity of minimizers of integral functional \[ I(u)= \int_{\Omega }F(x,u(x), Du) \,dx \] where \(u: \Omega \rightarrow \mathbb R^N\) and \(F\) is a quasi-convex function in the Morrey's sense. Let \(u\) be a minimizer and denote the singular set of \(u\) with \(\Sigma_u= \Omega-\Omega_u\), where \(\Omega_u\) is the set of \(x \in \Omega\) such that \(u \in C^{1,\sigma}(A,\mathbb R^N)\) for some \(\sigma>0\) and \(A\) is a neighborhood of \(x\). Under suitable assumptions of \(F\) and if \(u \in W^{1, \infty}(\Omega, \mathbb R^N)\) the authors give a precise estimate of the dimension of \(\Sigma_u\) i.e. dim\(_{\mathcal{H}}(\Sigma_u)\leq n-\delta\), with \(\delta>0\) depending on the data. By using the notion and the property of the set porosity, the result gives that the singular set is uniformely porous.
scientific article; zbMATH DE number 5149167
scientific article