6533b86ffe1ef96bd12ce866
RESEARCH PRODUCT
Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces
Renjin JiangRenjin Jiangsubject
Sobolev inequalityMathematics::Analysis of PDEsHölder conditionPoincaré inequality31C25 31C45 35B33 35B65Poisson equationSpace (mathematics)01 natural sciencesMeasure (mathematics)Sobolev inequalitysymbols.namesakeMathematics - Analysis of PDEs0103 physical sciencesFOS: Mathematics0101 mathematicsMathematicsMoser–Trudinger inequalityCurvatureRegular measureta111010102 general mathematicsMathematical analysisPoincaré inequalityMetric (mathematics)Riesz potentialsymbols010307 mathematical physicsPoisson's equationAnalysisAnalysis of PDEs (math.AP)description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2011-12-01 |