Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

Details Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

2011-01-05

arxiv.org/abs/1101.1016v2

Mathematics

Analysis of PDEs

Journal of Functional Analysis, to appear

Scientific paper

Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\mu$, where $Q>1$. Suppose that $(X,d,\mu)$ supports a (local) $(1,2)$-Poincar\'e inequality and a suitable curvature lower bound. For the Poisson equation $\Delta u=f$ on $(X,d,\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\"older continuity with optimal exponent of solutions is obtained.

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