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

TL;DR

This paper establishes higher differentiability of minimizers for a class of complex, anisotropic integral functionals with non-autonomous coefficients, advancing understanding of regularity in degenerate variational problems.

Contribution

It introduces novel techniques to prove higher differentiability for minimizers of non-autonomous, anisotropic functionals depending on the solution, a significant extension of existing regularity theory.

Findings

Higher differentiability of minimizers proven

Handles non-autonomous, anisotropic functionals

Extends regularity results to more complex 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.