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Resistance Conditions and Applications

100%

Kinnunen J., Silvestre P.

Analysis and Geometry in Metric Spaces

2013

tom 1

276-294

This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.

Stability and Continuity of Functions of Least Gradient

64%

Hakkarainen H., Korte R., Lahti P., Shanmugalingam N.

Analysis and Geometry in Metric Spaces

2015

tom 3

nr 1

In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.

Ograniczanie wyników

2 Analysis and Geometry in Metric Spaces

1 Hakkarainen H.

1 Kinnunen J.

1 Korte R.

1 Lahti P.

1 Shanmugalingam N.

1 Silvestre P.

1 2015

1 2013

Wyniki wyszukiwania

Resistance Conditions and Applications

Stability and Continuity of Functions of Least Gradient