Global geometric estimates for the heat equation via duality methods

TL;DR

This paper introduces a global, nonlinear approach using duality and Bernstein methods to analyze smoothing effects and geometric property conservation in solutions to the classical heat equation, applicable to various domain geometries.

Contribution

It develops a novel global framework combining duality and Bernstein techniques to study regularization and geometric invariance in heat flow, extending beyond maximum principle methods.

Findings

Establishes global smoothing estimates for the heat equation.

Proves conservation of geometric properties under heat flow.

Applicable to bounded and unbounded convex domains with Neumann conditions.

Abstract

We discuss first-order and second-order regularization effects for solutions to the classical heat equation. In particular we propose a global approach to study smoothing effects of Hamilton-Li-Yau type: such approach is nonlinear in spirit and it is based on the Bernstein method and duality techniques \`a la Evans. In a similar way, we also deal with the conservation of geometric properties for the heat flow as initiated by Brascamp-Lieb. In contrast to maximum principle methods based on sup-norm procedures, the integral method we adopt relies on contractivity properties for advection-diffusion equations and it applies to problems with homogeneous Neumann conditions posed equally on bounded and unbounded convex domains under suitable assumptions on their geometry.