Search results for " Hardy inequalities"

showing 3 items of 3 documents

Fractional Hardy inequalities and visibility of the boundary

2013

We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection to the boundedness of extension operators for fractional Sobolev spaces.

visibility of the boundaryPure mathematicsMathematics::Functional AnalysisInequalityfractional Hardy inequalitiesGeneral Mathematicsmedia_common.quotation_subject010102 general mathematicsVisibility (geometry)46E35 (26D15)Open setMathematics::Classical Analysis and ODEsOrder (ring theory)Boundary (topology)01 natural sciences010101 applied mathematicsMathematics - Classical Analysis and ODEsClassical Analysis and ODEs (math.CA)FOS: Mathematics0101 mathematicsMathematicsExposition (narrative)media_common
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Hardy inequalities and Assouad dimensions

2017

We establish both sufficient and necessary conditions for weighted Hardy inequalities in metric spaces in terms of Assouad (co)dimensions. Our sufficient conditions in the case where the complement is thin are new even in Euclidean spaces, while in the case of a thick complement we give new formulations for previously known sufficient conditions which reveal a natural duality between these two cases. Our necessary conditions are rather straight-forward generalizations from the unweighted case, but together with some examples they indicate the essential sharpness of our results. In addition, we consider the mixed case where the complement may contain both thick and thin parts.

Pure mathematics26D15 (Primary) 31E05 46E35 (Secondary)Partial differential equationGeneral Mathematics010102 general mathematicsDuality (mathematics)01 natural sciencesFunctional Analysis (math.FA)010101 applied mathematicsMathematics - Functional AnalysisMetric spaceAssouad (co)dimensionsMathematics - Classical Analysis and ODEsEuclidean geometryClassical Analysis and ODEs (math.CA)FOS: Mathematicsmetric spaces Hardy inequalities0101 mathematicsAnalysisMathematicsComplement (set theory)
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Limits of Sobolev homeomorphisms

2017

Let X; Y subset of R-2 be topologically equivalent bounded Lipschitz domains. We prove that weak and strong limits of homeomorphisms h: X (onto)-> Y in the Sobolev space W-1,W-p (X, R-2), p >= 2; are the same. As an application, we establish the existence of 2D-traction free minimal deformations for fairly general energy integrals. Peer reviewed

DIRICHLET ENERGYGeneral MathematicsDEFORMATIONSMONOTONE MAPPINGSLAPLACE EQUATION01 natural sciencesvariational integralsSobolev inequalityp-harmonic equationNONLINEAR ELASTICITYharmonic mappings111 MathematicsPOINTWISE HARDY INEQUALITIESREGULARITYSPACE0101 mathematicsMathematicsDISTORTIONSURFACESApplied Mathematics010102 general mathematicsMathematical analysisEnergy-minimal deformationsDirichlet's energy010101 applied mathematicsSobolev spaceapproximation of Sobolev homeomorphismsNonlinear elasticityJournal of the European Mathematical Society
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