Remarks about the mean value property and some weighted Poincar\'e-type inequalities

TL;DR

This paper establishes a quantitative stability theorem for harmonic functions in punctured domains, introduces new weighted Poincaré inequalities for vector fields, and extends mean value properties to harmonic functions in cones, linking these to rigidity results.

Contribution

It provides new stability results for overdetermined harmonic problems, introduces weighted Poincaré inequalities for vector fields, and extends mean value properties to harmonic functions in cones, with a duality relation to boundary value problems.

Findings

Quantitative stability theorem for harmonic functions in punctured domains.

New weighted Poincaré inequalities for vector fields.

Mean value property and Poincaré inequality for harmonic functions in cones.

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 Poincar\'e-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 Poincar\'e-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.