Search results for "Lipschitz regularity"

showing 2 items of 2 documents

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

2003

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.

Pure mathematicsMathematical analysisLipschitz continuityModulus of continuityCheeger-harmonicConvex metric spaceUniform continuityMetric spaceLipschitz domainPoincaré inequalityheat kerneldoubling measureMetric mapLipschitz regularitylogarithmic Sobolev inequalityMetric differentialhypercontractivityAnalysisNewtonian spaceMathematicsJournal of Functional Analysis
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On the Cauchy problem for microlocally symmetrizable hyperbolic systems with log-Lipschitz coefficients

2017

International audience; The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in H ∞ , if the coefficients enjoy enough smoothness in x. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysis is devoted to give an appropriate sense to the Cauchy problem, which is not evide…

Pure mathematicsloss of derivativeshyperbolic equationGeneral MathematicsMathematics::Analysis of PDEsmicrolocal symmetrizabilityhyperbolic equations; hyperbolic systems; log-lipschitz coefficientsSpace (mathematics)01 natural sciencesMathematics - Analysis of PDEslog-Lipschitz regularity; loss of derivatives; global and local Cauchy problem; well-posedness; non-characteristic Cauchy problemwell-posednessFOS: MathematicsInitial value problem[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Uniqueness0101 mathematics[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]MathematicsSmoothness (probability theory)Spacetimelog-lipschitz coefficients010102 general mathematicsglobal and local Cauchy problemExtension (predicate logic)Lipschitz continuitynon-characteristic Cauchy problemhyperbolic equationshyperbolic systemMathematics Subject Classificationlog-Lipschitz regularityhyperbolic systemsAnalysis of PDEs (math.AP)
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