Optimal Hölder regularity for solutions to Signorini-type obstacle problems

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 Hölder 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.