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Greenian potentials and concavity. (English) Zbl 0584.31003
Let D be a bounded domain in \({\mathbb{R}}^ n\), \(v\geq 0\) be a continuous function in D and \(G_ v\) be the Green operator relative to \(- (1/2)\Delta +v\). The author proves that the corresponding Green potentials satisfy a Brunn-Minkowski type inequality and obtains the following result:
If D is bounded convex and if \(f: D\to]0,\infty [\) and \(v^{-1/2}\) are concave (v\(\equiv 0\) or \(v>0)\), then \((G_ v f^ p)^{1/(2+p)}\) is concave, \(0\leq p\leq 1\).
Reviewer: V.Anandam

MSC:
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31A10 Integral representations, integral operators, integral equations methods in two dimensions
31B99 Higher-dimensional potential theory
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