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arXiv · 1604.04392

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

Daniel Wachsmuth
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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.

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Daniel Wachsmuth. 2016-04-15. 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. https://arxiv.org/abs/1604.04392

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