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Stability and Continuity of Functions of Least Gradient

H. Hakkarainen, R. Korte, P. Lahti, N. Shanmugalingam (2015)

Analysis and Geometry in Metric Spaces

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.

Strong Fubini properties for measure and category

Krzysztof Ciesielski, Miklós Laczkovich (2003)

Fundamenta Mathematicae

Let (FP) abbreviate the statement that ∫ 0 1 ( ∫ 0 1 f d y ) d x = ∫ 0 1 ( ∫ 0 1 f d x ) d y holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties...

Structural Properties of Solutions to Total Variation Regularization Problems

Wolfgang Ring (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere" , provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.

Subanalytic version of Whitney's extension theorem

Krzysztof Kurdyka, Wiesław Pawłucki (1997)

Studia Mathematica

For any subanalytic C k -Whitney field (k finite), we construct its subanalytic C k -extension to ℝ n . Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.

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