Higher Differentiability of Minimizers for Non-Autonomous Orthotropic Functionals

Abstract

We establish the higher differentiability for the minimizers of the following non-autonomous integral functionals \begin{equation*} \mathcal{F}(u,\Omega):= \, \int_\Omega \sum_{i=1}^{n} \, a_i(x) \lvert u_{x_i} \rvert^{p_i} dx, \end{equation*} with exponents $p_i \geq 2$ and with coefficients $a_i(x)$ that satisfy a suitable Sobolev regularity. The main result is obtained, as usual, by imposing a gap bound on the exponents $p_i$, which depends on the dimension and on the degree of regularity of the coefficients $a_i(x)$