Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth
The paper deals with the regularity of the local minimizers of non-autonomous integral functionals of the form \[ \int_\Omega \Phi^{p(x)}\left(\left(A^{\alpha\, \beta}_{i\,j}(x,u)D_iu^{\alpha}\,D_ju^{\beta}\right)^{\frac{1}{2}}\right) dx, \] where \(\Phi\) is an Orlicz function satisfying both the \(\Delta_2\) and \(\nabla_2\) conditions, \(p(x)\) is continuous and the function \(A(x,s)=\left( A^{\alpha\, \beta}_{i\,j}(x,s)\right) \) is uniformly continuous. Under suitable assumptions on the functions \(\Phi\) and \(p(x)\) the authors prove if \(u \in W^{1,p(x)}( \Omega)\) is a minimizer then \(u \in C^{0,\nu}( \Omega_0)\), for any \(\nu \in (0,1)\) where \( \Omega_0 \subset \Omega\) is an open subset with \(\dim_{\mathcal{H}}(\Omega- \Omega_0)<n-mp\), where \(m= \min_{\Omega}\,p\). Moreover, if the coefficients \(A^{\alpha\, \beta}_{i\,j}\) do not depend on \(u\), then \(\Omega_0=\Omega\).
scientific article; zbMATH DE number 6824798
scientific article