Rigidity of balls in the solid mean value property for polyharmonic functions
Nicola Abatangelo
TL;DR
This paper proves that in the context of polyharmonic functions, only balls satisfy the mean value property, and it offers a quantitative version of this uniqueness result.
Contribution
It establishes the uniqueness of balls for the mean value property in polyharmonic functions and adapts Kuran's harmonic function argument to this setting.
Findings
Balls are the only domains with the mean value property for polyharmonic functions.
A quantitative version of the mean value property characterization is provided.
The method adapts harmonic function techniques to polyharmonic functions.
Abstract
We show that balls are the only open bounded domains for which the mean value formula for polyharmonic functions holds. We do so by adapting an argument of \"U. Kuran for harmonic functions. Also, we provide a quantitative version of the same result.
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Taxonomy
TopicsHolomorphic and Operator Theory · Nonlinear Partial Differential Equations · Analytic and geometric function theory