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

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.