Accepted Paper

Inserted: 31 jul 2026
Last Updated: 31 jul 2026

Journal: Nonlinear Analysis, Theory, Methods and Applications
Volume: 194
Number: 111434
Year: 2020
Doi: 10.1016/j.na.2019.01.024

Abstract:

In this paper we prove the higher differentiability in the scale of Besov spaces of the solutions to a class of obstacle problems of the type $$\min\left\{\int_\Omega F(x, z, Dz): z\in \mathcal{K}_{\psi}(\Omega)\right\}.$$ Here $\Omega$ is an open bounded set of $\mathbb{R}^n$, $n \ge 2$, $\psi$ is a fixed function called {\it obstacle} and $\mathcal{K}_{\psi}(\Omega)$ is set of admissible functions $z \in W^{1,p}(\Omega)$ such that $z \ge \psi$ a.e. in $\Omega$. We assume that the gradient of the obstacle belongs to a suitable Besov space. The main novelty here is that we are not assuming any differentiability on the partial maps $x\mapsto F(x,z, Dz)$ and $z\mapsto F(x,z, Dz)$, but only their H\"older continuity.

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