Geodesic
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
Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 3, pp. 327-332.

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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.
DOI : 10.14712/1213-7243.2015.168
Classification : 35K10, 46E35
Keywords: Bochner integrable function; projection onto non-negative functions; parabolic equation 
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Wachsmuth, Daniel. 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. Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 3, pp. 327-332. doi : 10.14712/1213-7243.2015.168. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.168/

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