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Minkowski Inequality and nonlinear potential theory (part 1) Mazzieri, Lorenzo


In this talk, we first recall how some monotonicity formulas can be derived along the level set flow of the capacitary potential associated with a given bounded domain $\Omega$. A careful analysis is required in order to preserve the monotonicity across the singular times, leading in turn to a new quantitative version of the Willmore inequality. Remarkably, such analysis can be carried out without any \emph{a priori} knowledge of the size of the singular set. Hence, the same order of ideas applies to the $p$-capacitary potential of $\Omega$, whose critical set, for $p\neq2$, is not necessarily negligible. In this context, a generalised version of the Minkowski inequality is deduced. Joint works with M. Fogagnolo and L. Mazzieri.

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