Search results for "31c45"

showing 3 items of 13 documents

Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces

2015

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Pure mathematicsWolff potentialScale (ratio)Weak Lebesgue spaceVariable exponentMathematics::Classical Analysis and ODEsLebesgue's number lemmaNon-standard growth conditionIntegrability of solutionssymbols.namesakeMathematics - Analysis of PDEsReal interpolationFOS: MathematicsLp spaceMathematicsLaplace's equationMathematics::Functional AnalysisVariable exponentIntegrability estimatesRiesz potentialApplied MathematicsMathematical analysisFunctional Analysis (math.FA)Mathematics - Functional AnalysissymbolsRiesz potential47H99 (Primary) 46B70 46E30 35J60 31C45 (Secondary)Analysis of PDEs (math.AP)
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Existence and almost uniqueness for p -harmonic Green functions on bounded domains in metric spaces

2020

We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted $\mathbf{R}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various c…

Pure mathematicsCapacitary potential; Doubling measure; Metric space; p-harmonic Green function; Poincar? inequality; Singular function31C45 (Primary) 30L99 31C15 31E05 35J92 49Q20 (Secondary)Harmonic (mathematics)Mathematical Analysis01 natural sciencesMeasure (mathematics)Domain (mathematical analysis)Mathematics - Analysis of PDEscapacitary potentialMatematisk analysFOS: MathematicsUniqueness0101 mathematicsMathematicsComplement (set theory)p-harmonicApplied Mathematics010102 general mathematicsmetric spacemetriset avaruudet010101 applied mathematicsMetric spacePoincaré inequalityBounded functionMetric (mathematics)doubling measurepotentiaaliteoriasingular functiongreen functionAnalysisAnalysis of PDEs (math.AP)
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Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

2011

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.

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)
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