Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

TL;DR

This paper establishes higher regularity results for bounded solutions to obstacle problems with non-standard growth, under conditions that do not depend on the dimension, broadening the understanding of solution smoothness in complex variational problems.

Contribution

It provides dimension-independent regularity results for obstacle problems with non-standard growth, including cases with Sobolev coefficients below the critical threshold.

Findings

Higher differentiability for bounded solutions achieved.

Dimension-free conditions on growth and ellipticity gaps established.

Regularity results applicable to coefficients in sub-critical Sobolev spaces.

Abstract

In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_{\Omega}\left<A(x, Du), D(\varphi-u)\right> \, \dx\ge0\qquad\forall %\varphi\in W^{1,q}(\Omega) \quad {\mathrm{s.t.}} \quad \varphi \ge \psi %\end{equation*} \begin{equation*} \min \left\{\int_{\Omega}f(x, Dv)\, \dx\,:\, v\in \mathcal{K}_\psi(\Omega)\right\} \end{equation*} where $\psi$ is the obstacle, $\mathcal{K}_\psi(\Omega)=\{v\in u_0+W^{1, p}_{0}(\Omega, \R): v\ge\psi \text{ a.e. in }\Omega\}$, with $u_0 \in W^{1,p}(\Omega)$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x, Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $x\mapsto A(x, \xi)$ is independent of the dimension $n$ and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $W^{1,n}$.