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 $F(u,\Omega )≔{\int }_{\Omega }\sum _{i=1}^n{a}_i\left(x\right){\left|u_{x_i}\right|}^{p_i}dx,$with exponents $p_i\ge 2$ and with coefficients $a_i\left(x\right)$ 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 $a_i\left(x\right)$.