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Optimal Hölder regularity for solutions to Signorini-type obstacle problems

Ki-Ahm Lee, Se-Chan Lee, Waldemar Schefer

TL;DR

The paper establishes optimal Hölder regularity for Signorini-type obstacle problems where the obstacle acts only on a codimension-one subset, by employing a capacity density condition on the contact set and a combination of Alt–Caffarelli–Friedman and Almgren monotonicity tools. It develops a robust framework linking Signorini-type problems to mixed boundary value problems, leveraging Maz'ya’s inequality to derive $C^{1/2}$-regularity and, in special cases, almost-optimal $C^{1/2-\varepsilon}$ regularity. In two dimensions, it provides a detailed blow-up analysis and a full classification of limiting profiles, concluding $C^{1/2}$ regularity for half-line obstacles with explicit blow-up forms. The work also extends to irregular obstacles with jump-type discontinuities, showing that the solution behavior near the jump is governed by the dominating obstacle and inherits the optimal regularity. Overall, the results connect potential-theoretic capacity concepts with free-boundary techniques to characterize precise regularity thresholds for generalized 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 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.

Optimal Hölder regularity for solutions to Signorini-type obstacle problems

TL;DR

The paper establishes optimal Hölder regularity for Signorini-type obstacle problems where the obstacle acts only on a codimension-one subset, by employing a capacity density condition on the contact set and a combination of Alt–Caffarelli–Friedman and Almgren monotonicity tools. It develops a robust framework linking Signorini-type problems to mixed boundary value problems, leveraging Maz'ya’s inequality to derive -regularity and, in special cases, almost-optimal regularity. In two dimensions, it provides a detailed blow-up analysis and a full classification of limiting profiles, concluding regularity for half-line obstacles with explicit blow-up forms. The work also extends to irregular obstacles with jump-type discontinuities, showing that the solution behavior near the jump is governed by the dominating obstacle and inherits the optimal regularity. Overall, the results connect potential-theoretic capacity concepts with free-boundary techniques to characterize precise regularity thresholds for generalized 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 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.

Paper Structure

This paper contains 24 sections, 38 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Historical background
  4. Strategy of the proofs
  5. Overview
  6. The general obstacle problem
  7. Existence and uniqueness
  8. Continuity and harmonicity of solutions
  9. Hölder and Lipschitz regularity
  10. Optimal Hölder regularity for Signorini-type problems under a capacity density condition
  11. Alt--Caffarelli--Friedman-type frequency function
  12. Proof of \ref{['thm:optimalreg']}
  13. Generalization for non-zero obstacles
  14. Almost optimal Hölder regularity in the half-hyperplane obstacle case
  15. Well posedness of Mixed BVPs
  16. ...and 9 more sections

Key Result

Theorem 2.3

Let $\Omega\subset\mathbb R^n$ be open and bounded. Suppose that $F\subset \Omega$, $g\in H^1(\Omega)$, and $\psi: F \to \overline{\mathbb R}$. If $\mathcal{K}_{g, \psi, F}(\Omega)$ is not empty, then there exists a unique solution to the $\mathcal{K}_{g, \psi, F}(\Omega)$-obstacle problem.

Figures (1)

  • Figure 1: The unit ball with cone $F_\alpha$.

Theorems & Definitions (82)

  • Definition 2.1: General obstacle problems
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4: Adams' criterion, BB11
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • Proposition 2.8
  • ...and 72 more