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Inverse mean curvature flow with outer obstacle

Kai Xu

TL;DR

This work develops a rigorous variational framework for the inverse mean curvature flow (IMCF) in bounded domains under an outer obstacle boundary condition, where flowing hypersurfaces stick tangentially to the boundary. It constructs and analyzes weak IMCF solutions using elliptic regularization, calibration, and energy methods, establishing existence, uniqueness (via maximality), and $C^{1,\alpha}$ regularity up to the obstacle. A key novelty is a robust outer-obstacle formulation that unifies sublevel-set and Dirichlet-energy perspectives and yields well-posed initial-value problems; this is complemented by Liouville-type results on the half-space and parabolic estimates near smooth obstacles, which underpin the blow-up analysis and regularity theory. The results provide a canonical way to obtain nontrivial maximal weak solutions in bounded domains and relate to the global behavior of IMCF through isoperimetric considerations, with potential implications for Hawking mass monotonicity and scalar curvature problems. Overall, the paper advances the analytic toolbox for weak IMCF with boundary constraints and clarifies the interaction between obstacle geometry, calibration, and parabolic regularity in the evolution.

Abstract

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish $C^{1,α}$ regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.

Inverse mean curvature flow with outer obstacle

TL;DR

This work develops a rigorous variational framework for the inverse mean curvature flow (IMCF) in bounded domains under an outer obstacle boundary condition, where flowing hypersurfaces stick tangentially to the boundary. It constructs and analyzes weak IMCF solutions using elliptic regularization, calibration, and energy methods, establishing existence, uniqueness (via maximality), and regularity up to the obstacle. A key novelty is a robust outer-obstacle formulation that unifies sublevel-set and Dirichlet-energy perspectives and yields well-posed initial-value problems; this is complemented by Liouville-type results on the half-space and parabolic estimates near smooth obstacles, which underpin the blow-up analysis and regularity theory. The results provide a canonical way to obtain nontrivial maximal weak solutions in bounded domains and relate to the global behavior of IMCF through isoperimetric considerations, with potential implications for Hawking mass monotonicity and scalar curvature problems. Overall, the paper advances the analytic toolbox for weak IMCF with boundary constraints and clarifies the interaction between obstacle geometry, calibration, and parabolic regularity in the evolution.

Abstract

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.
Paper Structure (26 sections, 54 theorems, 320 equations, 6 figures)

This paper contains 26 sections, 54 theorems, 320 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Smooth and interior weak solutions
  5. Weak solutions with weights
  6. Calibrated weak solutions
  7. The elliptic regularization process
  8. An excess inequality
  9. Further analytic properties of weak solutions
  10. Weak formulations with outer obstacle
  11. Formulation using sub-level sets
  12. Outward minimizing properties
  13. Formulation using 1-Dirichlet energy
  14. Boundary orthogonality of calibration
  15. A weak maximum principle
  16. ...and 11 more sections

Key Result

Lemma 1.6

Let $\Omega\subset M$ be a locally Lipschitz domain, and $u$ be a weak solution of IMCF in $\Omega$ which is calibrated by $\nu$. If $\nu\cdot\nu_\Omega=1$ on $\partial\Omega$ in the trace sense, then the solution $u$ respects the outer obstacle $\partial\Omega$.

Figures (6)

  • Figure 1: translating cycloids.
  • Figure 2: Expanding nephroids.
  • Figure 3: IMCF with obstacle in a punctured ellipse
  • Figure 4: an example of disconnected sub-level set.
  • Figure 5: translating cycloids in a strip region. The bold curve represents $\partial E_t$.
  • ...and 1 more figures

Theorems & Definitions (127)

  • Example 1.1
  • Example 1.2: translating cycloids
  • Example 1.3: expanding nephroid
  • Example 1.4
  • Lemma 1.6: identical with Lemma \ref{['lemma-obs:bd_orthogonal']}
  • Theorem 1.7: concise version of Theorem \ref{['thm-exist:main']}
  • Theorem 1.8: identical with Theorem \ref{['thm-max:existence']}
  • Theorem 1.9: identical with Theorem \ref{['thm-halfplane:liouville']}
  • Remark 2.1
  • Remark 2.2
  • ...and 117 more