Optimal H\"{o}lder regularity for solutions to Signorini-type obstacle problems

TL;DR

This paper establishes the optimal Hölder regularity for solutions to Signorini-type obstacle problems, where the obstacle is imposed on a subset of the domain, using advanced potential theory and monotonicity formulas.

Contribution

It introduces new regularity results for Signorini problems with obstacles on lower-dimensional subsets, extending classical obstacle problem theory.

Findings

Proves optimal Hölder regularity for Signorini-type problems.

Develops new techniques using capacities and monotonicity formulas.

Applies results to irregular obstacle problems.

Abstract

We study the existence, uniqueness, and regularity of weak solutions to a class of obstacle problems, where the obstacle condition can be imposed on a subset of the domain. In particular, we establish the optimal H\"older regularity for Signorini-type problems, that is, the obstacle condition is imposed only on a subset of codimension one. For this purpose, we employ capacities, Alt--Caffarelli--Friedman-type and Almgren-type monotonicity formulae, and investigate an associated mixed boundary value problem. Further, we apply this problem to study classical obstacle problems for irregular obstacles.