Evolution Equations on Manifolds with Conical Singularities

2025/06/20 by Schrohe, Elmar
#35B40 #35C20 #35K59 #35K65 #35R01 #58J40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2506.17481

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_∞-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.

Citations

• Asymptotics of solutions to the porous medium equation near conical singularities
• Phase transitions and minimal interfaces on manifolds with conical singularities
• A Short Introduction to the Analysis on Manifolds with Conical Singularities
• The Fractional Porous Medium Equation on noncompact Riemannian manifolds
• The porous medium equation with measure data on negatively curved\n Riemannian manifolds
• Fundamental solution and long time behaviour of the Porous Medium Equation in Hyperbolic Space
• Ricci flow on asymptotically conical surfaces with nontrivial topology
• The Resolvent of Closed Extensions of Cone Differential Operators
• Differential Operators on Conic Manifolds: Maximal Regularity and Parabolic Equations

Related

• Asymptotics of solutions to the porous medium equation near conical singularities
• A Short Introduction to the Analysis on Manifolds with Conical Singularities
• Evolution equations in Riemannian geometry
• Maximal L^q-regularity for the Laplacian on manifolds with edges
• Conformal Laplacian and Conical Singularities