The regularity of the positive part of functions in $L^2(I; H^1(\Omega)) \cap H^1(I; H^1(\Omega)^*)$ with applications to parabolic equations

TL;DR

This paper investigates the regularity properties of the positive part of functions in certain Sobolev spaces and demonstrates their implications for the analysis of weak solutions to parabolic equations.

Contribution

It provides a counter-example showing reduced regularity of the positive part and establishes an integration-by-parts formula useful for parabolic PDEs.

Findings

The positive part of functions in the specified space may lack certain regularity.

An integration-by-parts formula for the positive part is proved.

Application to non-negativity of weak solutions in parabolic equations.

Abstract

Let $u\in L^2(I; H^1(\Omega))$ with $\partial_t u\in L^2(I; H^1(\Omega)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \not\in L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.