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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.

Sur une intégrale double

Gérassime Orlow (1882)

Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale

The area formula for W 1 , n -mappings

Jan Malý (1994)

Commentationes Mathematicae Universitatis Carolinae

Let f be a mapping in the Sobolev space W 1 , n ( Ω , 𝐑 n ) . Then the change of variables, or area formula holds for f provided removing from counting into the multiplicity function the set where f is not approximately Hölder continuous. This exceptional set has Hausdorff dimension zero.

Visualisation of the electromagnetic vector fields

Bartoň, Stanislav (2023)

Programs and Algorithms of Numerical Mathematics

Modern computer algebra software can be used to visualize vector fields. One of the most used is the Maple program. This program is used to visualize two and three-dimensional vector fields. The possibilities of plotting direction vectors, lines of force, equipotential curves and the method of colouring the surface area for two-dimensional cases are shown step by step. For three-dimensional arrays, these methods are applied to various slices of three-dimensional space, such as a plane or a cylindrical...

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