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The M-components of level sets of continuous functions in WBV.

Coloma Ballester, Vicent Caselles (2001)

Publicacions Matemàtiques

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and...

Transport equations with partially B V velocities

Nicolas Lerner (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ∂ t u + X u = f , u | t = 0 = g , where X is the vector fieldwith a boundedness condition on the divergence of each vector field a 1 , a 2 . This model was studied in the paper [LL] with a W 1 , 1 regularity assumption replacing our B V hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of...

Uniformly bounded composition operators in the banach space of bounded (p, k)-variation in the sense of Riesz-Popoviciu

Francy Armao, Dorota Głazowska, Sergio Rivas, Jessica Rojas (2013)

Open Mathematics

We prove that if the composition operator F generated by a function f: [a, b] × ℝ → ℝ maps the space of bounded (p, k)-variation in the sense of Riesz-Popoviciu, p ≥ 1, k an integer, denoted by RV(p,k)[a, b], into itself and is uniformly bounded then RV(p,k)[a, b] satisfies the Matkowski condition.

Variation of quasiconformal mappings on lines

Leonid V. Kovalev, Jani Onninen (2009)

Studia Mathematica

We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Hölder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p > 1 but not necessarily for p = 1.

Variational measures related to local systems and the Ward property of 𝒫 -adic path bases

Donatella Bongiorno, Luisa Di Piazza, Valentin A. Skvortsov (2006)

Czechoslovak Mathematical Journal

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a 𝒫 -adic path system that defines a differentiation basis which does not possess Ward property.

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