Sharp estimates for the gradient of solutions to the heat equation

TL;DR

This paper derives sharp pointwise estimates for the gradient of solutions to the heat equation under Dirichlet and Neumann boundary conditions in a half-space, using optimization techniques over Lebesgue space data.

Contribution

It introduces new sharp estimates for the heat equation gradient with explicit coefficients derived from optimization problems.

Findings

Sharp gradient estimates for heat equation solutions

Explicit coefficients obtained via optimization methods

Applicable to boundary conditions in half-spaces

Abstract

Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.