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Evolution Equations on Manifolds with Conical Singularities

Elmar Schrohe

TL;DR

This work develops a robust framework for nonlinear evolution on manifolds with conical singularities by combining cone-geometry–aware function spaces with maximal regularity and bounded $H_\infty$-calculus. It analyzes closed extensions of cone differential operators, especially the cone Laplacian, to ensure well-posedness of semilinear parabolic problems via the Clément–Li theorem, and demonstrates the approach on concrete PDEs: the porous medium equation, its fractional variant, Yamabe flow, and the Cahn–Hilliard equation. Key contributions include explicit descriptions of minimal/maximal domains, Mellin-symbol–driven asymptotics, interpolation spaces for initial data, and perturbation arguments that yield short-time and, in some cases, global existence and attractors. The results reveal how geometry near conical tips governs solution behavior and asymptotics through spectral data of cross-sections, providing tools for geometric evolution and diffusion on singular spaces. Overall, the article delivers a comprehensive analytic toolkit for nonlinear diffusion and phase-field equations on manifolds with conical singularities with potential geometric-flow applications.

Abstract

This is an introduction to the analysis of nonlinear evolution equations on manifolds with conical singularities via maximal regularity techniques. We address the specific difficulties due to the singularities, in particular the choice of extensions of the conic Laplacian that guarantee the existence of a bounded $H_\infty$-calculus. We introduce the relevant technical tools and survey, as main examples, applications to the porous medium equation, the fractional porous medium equation, the Yamabe flow, and the Cahn-Hilliard equation.

Evolution Equations on Manifolds with Conical Singularities

TL;DR

This work develops a robust framework for nonlinear evolution on manifolds with conical singularities by combining cone-geometry–aware function spaces with maximal regularity and bounded -calculus. It analyzes closed extensions of cone differential operators, especially the cone Laplacian, to ensure well-posedness of semilinear parabolic problems via the Clément–Li theorem, and demonstrates the approach on concrete PDEs: the porous medium equation, its fractional variant, Yamabe flow, and the Cahn–Hilliard equation. Key contributions include explicit descriptions of minimal/maximal domains, Mellin-symbol–driven asymptotics, interpolation spaces for initial data, and perturbation arguments that yield short-time and, in some cases, global existence and attractors. The results reveal how geometry near conical tips governs solution behavior and asymptotics through spectral data of cross-sections, providing tools for geometric evolution and diffusion on singular spaces. Overall, the article delivers a comprehensive analytic toolkit for nonlinear diffusion and phase-field equations on manifolds with conical singularities with potential geometric-flow applications.

Abstract

This is an introduction to the analysis of nonlinear evolution equations on manifolds with conical singularities via maximal regularity techniques. We address the specific difficulties due to the singularities, in particular the choice of extensions of the conic Laplacian that guarantee the existence of a bounded -calculus. We introduce the relevant technical tools and survey, as main examples, applications to the porous medium equation, the fractional porous medium equation, the Yamabe flow, and the Cahn-Hilliard equation.

Paper Structure

This paper contains 19 sections, 33 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Conical Singularities and Cone Sobolev Spaces
  3. Manifolds with Conical Singularities
  4. Cone Differential Operators
  5. Cone Sobolev Spaces
  6. Closed Extensions of Cone Differential Operators
  7. The Model Cone Operator
  8. Closed Extensions
  9. Maximal Regularity, the Clément-Li Theorem and Applications
  10. The Laplace-Beltrami Operator
  11. Interpolation
  12. Example: The Porous Medium Equation
  13. Short Time Solutions
  14. Global Solutions
  15. Asymptotics near the Conical Points
  16. ...and 4 more sections

Key Result

Corollary 1

Let $1\le p<\infty$ and $s>(n+1)/p$, $\gamma\in \mathbb R$. Then a function $u\in \mathcal{H}^{s,\gamma}_p(\mathbb B)$ is continuous on $\mathop{\mathrm{int}}\nolimits \mathbb B$. Near $\partial \mathbb B$ it satisfies the estimate for some constant $c\ge0$. In particular, $u(x,y) \to 0$ as $x\to 0$ whenever $\gamma>(n+1)/2$.

Theorems & Definitions (66)

  • Remark 1
  • Definition 1
  • Example 1
  • Definition 2
  • Example 2
  • Definition 3
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • ...and 56 more