Thursday, March 4, 2021 — 2:00 pm, online talk

Regularity results for a class of obstacle problems

Abstract:  We present some higher differentiability results of integer and fractional order of the gradient of solutions to variational obstacle problems of the form

$$\min \left\{\int_\Omega F(x,u,Du)dx \: : \: u\in u_0+\mathcal{K}_\psi(\Omega) \right\},$$

where $\Omega\subset\mathbb{R}^n$ is a bounded set, $n\geq 2$. The boundary datum $u_0$ and the obstacle $\psi$ belong to the Sobolev class $W^{1,p}(\Omega)$ and the admissible class $\mathcal{K}_\psi(\Omega)$ is defined as follows

$$\mathcal{K}_\psi(\Omega)={v\in u_0+W^{1,p}(\Omega):\,\, v\geq \psi}.$$

The energy density $F$ is assumed to be convex and of class $C^2$ and satisfies $p$- growth condition, $p\geq 2$, with respect to the gradient variable.

We show that a Besov regularity assumption on the gradient of the obstacle $\psi$ transfer to the gradient of the solution.

The results are contained in joint works with Michela Eleuteri ([1], [2]).


[1] M. Eleuteri, A.Passarelli di Napoli. Higher differentiability for solutions to a class of obstacle problems. Calc. Var. and PDE’s (2018).

[2] M. Eleuteri, A.Passarelli di Napoli. Regularity results for a class of non-differentiable obstacle problems. Nonlinear Anal. 194 (2020).