Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces

TL;DR

This paper explores nonlinear diffusion equations on smooth metric measure spaces and their connections to geometric and functional inequalities, presenting new theoretical insights and results in this mathematical framework.

Contribution

It introduces new results linking nonlinear diffusion equations with geometric and functional inequalities on smooth metric measure spaces.

Findings

New inequalities established for nonlinear diffusion equations

Connections between diffusion processes and geometric properties clarified

Several theoretical results presented in a summer school style

Abstract

This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast types and their links with some geometric and functional inequalities in the framework of smooth metric measure spaces. The article presents some introduction in a summer school style as well as several new results.