Abstract
We prove a mean value formula for weak solutions of \(\hbox {div}(|y|^{a}\hbox {grad} u)=0\) in \({\mathbb {R}}^{n+1}=\{ (x,y):\ x\in {\mathbb {R}}^{n},\ y\in {\mathbb {R}} \}\), \(-1<a<1\), and balls centered at points of the form \((x,0)\). We obtain an explicit nonlocal kernel for the mean value formula for solutions of \((-\triangle )^{s}f=0\) on a domain \(D\) of \({\mathbb {R}}^{n}\). When \(D\) is Lipschitz, we prove a Besov type regularity improvement for the solutions of \((-\triangle )^{s}f=0\).
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The research was supported by CONICET, ANPCyT (MINCyT) and UNL.
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Communicated by Serguei Denissov.
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Aimar, H., Beltritti, G. & Gómez, I. Improvement of Besov Regularity for Solutions of the Fractional Laplacian. Constr Approx 41, 219–229 (2015). https://doi.org/10.1007/s00365-014-9256-0
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DOI: https://doi.org/10.1007/s00365-014-9256-0