Daniel Wachsmuth
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

Comment.Math.Univ.Carolin. 57,3 (2016) 327-332.

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 $\partial_t u^+ \notin L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Keywords: Bochner integrable function; projection onto non-negative functions; parabolic equation

DOI: DOI 10.14712/1213-7243.2015.168
AMS Subject Classification: 46E35 35K10

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