Explicit higher regularity on a Cauchy problem with mixed Neumann-power type boundary conditions

TL;DR

This paper studies the regularity of solutions to a Cauchy problem with mixed boundary conditions, establishing explicit gradient estimates in $L^p$ spaces and discussing steady-state regularity.

Contribution

It provides new explicit regularity estimates for weak solutions of a Cauchy problem with mixed Neumann-power boundary conditions, extending understanding of solution smoothness.

Findings

Existence of weak solutions with explicit $L^p$ gradient estimates

Regularity results for steady-state solutions

Analysis applicable for $p > 2$ in $L^p$ spaces

Abstract

We investigate the regularity in $L^p$ ($p>2$) of the gradient of any weak solution of a Cauchy problem with mixed Neumann-power type boundary conditions. Under suitable assumptions we prove the existence of weak solutions that satisfy explicit estimates. Some considerations on the steady-state regularity are discussed.