Regularity for minimizers of degenerate, non-autonomous, orthotropic integral functionals
Antonio Giuseppe Grimaldi, Stefania Russo
TL;DR
This work extends regularity theory for orthotropic variational functionals by treating a non-autonomous, degenerate density that depends on the solution and on $x$. Using a three-stage approach—regularization of coefficients, a priori estimates via a difference-quotient technique, and a careful approximation scheme—the authors prove higher differentiability of integer order for locally bounded minimizers: $V_{p_i}(u_{x_i}) \in W^{1,2}_{loc}$ for all $i$, under Sobolev regularity assumptions $a_i\in W^{1,r}_{loc}$ and $\omega\in W^{1,(\mathbf{p}+2)/(\mathbf{p}+1)}_{loc}$ with a forcing term $g\in L^r_{loc}$ and $r>p_n+2$. The paper provides explicit two-tier estimates: bounds on $\int_{B_{R/4}} |u_{x_i}|^{p_i+2}$ and on $\int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}$, highlighting how the data control the regularity. This advances the understanding of non-standard growth in anisotropic settings and offers a robust framework to handle forcing terms within the non-autonomous orthotropic context.
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 \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.