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,\Omega):= \,\sum_{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int_\Omega \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx- \int_\Omega \omega(x)u(x) dx, \end{equation*} where the exponents $ p_i \geq 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.