Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

TL;DR

This paper proves higher differentiability of solutions to non-autonomous obstacle problems with sub-quadratic growth, linking obstacle regularity to solution regularity under Sobolev or Besov conditions.

Contribution

It introduces new higher differentiability results for obstacle problems with p-growth where 1<p<2, under regularity assumptions on the obstacle and the integrand's coefficients.

Findings

Higher differentiability transfers from obstacle to solution under Sobolev/Besov regularity.

Results apply to sub-quadratic growth conditions with p between 1 and 2.

Regularity of the coefficient function a(x) influences solution smoothness.

Abstract

We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_{\Omega}f(x, Dv(x))\,:\, v\in \mathcal{K}_\psi(\Omega)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_\psi(\Omega)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(\Omega)$ such that $v\ge\psi$ a. e. in $\Omega$, where $u_0\in W^{1,p}(\Omega)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gradient of the obstacle $\psi$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_\xi f(x,\xi)$ belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. $f(x, \xi)\approx a(x)|\xi|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev or Besov-Lipschitz space.