Wednesday, May 5, 2021 — 4:00 pm — online talk

Abstract: We present some regularity properties of solutions to variational inequalities of the form
$$\int_\Omega \langle \mathcal A(x, u, Du), D(\varphi − u) \rangle \, dx \geq \int_\Omega \mathcal B(x, u, Du)(\varphi − u) \, dx, \quad \forall \varphi \in \mathcal K_\psi(\Omega).$$
Here $\Omega$ is a bounded open set of $\mathbb R^n$, $n \geq 2$, the function $\psi: \: \Omega \to [−\infty, +\infty)$, called obstacle, belongs to the Sobolev class $W^{1,p} (\Omega)$ and $\mathcal K_\psi(\Omega) =\{ \: w \in W^{1,p}(\Omega) \: : \: w \geq \psi \; \text{ q.o. in } \; \Omega \: \}$ is the class of the admissible functions.

First we establish a local Calderòn-Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces $L^{pq}$, for every $q \in (1, \infty)$, provided the partial map $(x, u) \mapsto \mathcal A(x, u, \xi)$ is Hölder continuous and $\mathcal B(x, u, \xi)$ satisfies a suitable growth condition.

Next, this estimate allows us to prove that a higher differentiability in the scale of Besov spaces of the gradient of the obstacle transfers to the gradient of the solutions.

References:
[1] A.G. Grimaldi. Regularity results for solutions to a class of obstacle problems. Preprint (2021)