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Free boundary regularity for semilinear variational problems with a topological constraint

Michael Novack, Daniel Restrepo, Anna Skorobogatova

Abstract

We study a class of semilinear free boundary problems in which admissible functions $u$ have a topological constraint, or spanning condition, on their 1-level set. This constraint forces $\{u=1\}$, which is the free boundary, to behave like a surface with some special types of singularities attached to a fixed boundary frame, in the spirit of the Plateau problem \cite{HP16}. Two such free boundary problems are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn formulation of the Plateau problem. We establish the existence of minimizers and study their regularity properties, obtaining the optimal Lipschitz regularity of minimizers and analytic regularity for the free boundaries away from a codimension two singular set. The singularity models for these problems are given by conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems.

Free boundary regularity for semilinear variational problems with a topological constraint

Abstract

We study a class of semilinear free boundary problems in which admissible functions have a topological constraint, or spanning condition, on their 1-level set. This constraint forces , which is the free boundary, to behave like a surface with some special types of singularities attached to a fixed boundary frame, in the spirit of the Plateau problem \cite{HP16}. Two such free boundary problems are the minimization of capacity among surfaces sharing a common boundary and an Allen-Cahn formulation of the Plateau problem. We establish the existence of minimizers and study their regularity properties, obtaining the optimal Lipschitz regularity of minimizers and analytic regularity for the free boundaries away from a codimension two singular set. The singularity models for these problems are given by conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems.

Paper Structure

This paper contains 24 sections, 50 theorems, 382 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Discussion
  5. Commentary on proofs and structure of article
  6. Further remarks
  7. Notation
  8. Acknowledgements
  9. Monotonicity properties and regularity of stationary solutions
  10. Statement of Theorem \ref{['thm:holder reg for crit points']} and application to proof of Theorem \ref{['thm:main regularity theorem']}(i)
  11. Monotonicity formulae
  12. A criterion for Hölder regularity
  13. Frequency bounds and Hölder regularity
  14. Compactness
  15. Tangent functions and unique continuation
  16. ...and 9 more sections

Key Result

Theorem 1.1

If $\mathbf{W} = \mathbb{R}^{n+1}\setminus \Omega$ is compact, $\mathcal{C}$ is a spanning class for $\mathbf{W}$ satisfying eq:spanning class assumption, $F$ and $V$ satisfy eq:A1-eq:A3, and $u$ is a minimizer of new model intro or new model volume constrained, then (i):$u$ is locally Lipschitz in where $\mathcal{R}(u)$ is locally an analytic $n$-dimensional manifold and $\mathcal{S}(u)$ is a cl

Figures (1)

  • Figure 1.1: Shown above are two different configurations of $\mathbf{W}\subset \mathbb{R}^2$, generators for an associated spanning class $\mathcal{C}$, and a spanning set. In both cases, $\mathbf{W}$ is the union of the gray balls, $\mathcal{C}$ is the family of smooth loops homotopic to some $\gamma_i$, and the example spanning sets are composed of line segments.

Theorems & Definitions (125)

  • Theorem 1.1: Regularity of minimizers
  • Theorem 1.2: Existence of minimizers
  • Remark 1.3: Optimality of the assumptions in Theorem \ref{['thm:main existence theorem']}
  • Remark 1.4: On the free boundary decomposition
  • Remark 1.5: Connection to optimal partition & segregation problems
  • Remark 1.6: Further analysis of singularities
  • Remark 1.7: Extension of existence from maggi2023hierarchy
  • Remark 1.8: Extension to bounded domains
  • Remark 2.1: Spanning and the Euler-Lagrange equations
  • Theorem 2.2: Lipschitz regularity for critical points with frequency lower bound
  • ...and 115 more