On a class of quasilinear operators on smooth metric measure spaces

TL;DR

This paper establishes sharp continuity estimates and eigenvalue bounds for a broad class of quasilinear parabolic equations on smooth metric measure spaces, utilizing one-dimensional comparisons for precise results.

Contribution

It introduces a novel approach to derive sharp estimates and bounds for quasilinear operators on metric measure spaces using one-dimensional operator comparisons.

Findings

Sharp modulus of continuity estimates for solutions

Optimal lower bounds for first Dirichlet eigenvalues

Applicable to non-variational quasilinear operators

Abstract

We derive sharp estimates on the modulus of continuity for solutions of a large class of quasilinear isotropic parabolic equations on smooth metric measure spaces (with Dirichlet or Neumann boundary condition in case the boundary is non-empty). We also derive optimal lower bounds for the first Dirichlet eigenvalue of a class of homogeneous quasilinear operators, which include non-variational operators. The main feature is that this class of operators have corresponding one-dimensional operators, which allow sharp comparisons with solutions of one-dimensional equations.