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arXiv · 2308.07000

Remarks about the mean value property and some weighted Poincaré-type inequalities

Giorgio Poggesi
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Abstract

We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established by Enciso and Peralta-Salas, and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincaré-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated by the author concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincaré-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result due to Payne and Schaefer.

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Giorgio Poggesi. 2023-08-21. Remarks about the mean value property and some weighted Poincaré-type inequalities. https://arxiv.org/abs/2308.07000

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