Regularity for minimizers of degenerate, non-autonomous, orthotropic integral functionals

Abstract

We prove the higher differentiability of integer order of locally bounded minimizers of integral functionals of the form \begin{equation*} \mathcal{F}(u,Ω):= \,\sum_{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int_Ω\, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx- \int_Ωω(x)u(x) dx, \end{equation*} where the exponents $p_i\ge 2$ and the coefficients $a_i(x)$ satisfy a suitable Sobolev regularity. The main novelty consists in dealing with non-autonomous, anisotropic functionals, which depend also on the solution.