Potential Analysis

, Volume 37, Issue 3, pp 281–301 | Cite as

Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces

Article

Abstract

Let (X, d) be a pathwise connected metric space equipped with an Ahlfors Q-regular measure μ, Q ∈ [1, ∞ ). Suppose that (X, d, μ) supports a 2-Poincaré inequality and a Sobolev–Poincaré type inequality for the corresponding “Gaussian measure”. The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation Δu = f, where \(f\in L^{p}_{\rm{loc}}\). When p > Q, the local Lipschitz continuity of u is established.

Keywords

Lipschitz regularity Poincaré inequality Newtonian space Heat kernel Poisson equation 

Mathematics Subject Classifications (2010)

31C25 31B05 35B05 35B45 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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