We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.
We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral.
This paper describes a compact perceptual image model intended for morphological representation of the visual information contained in natural images. We explain why the total variation can be a criterion to split the information between the two main visual structures, which are the sketch and the microtextures. We deduce a morphological decomposition scheme, based on a segmentation where the borders of the regions correspond to the location of the topological singularities of a topographic map. This...
We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.
In this note we prove compactness for the Cahn–Hilliard functional without assuming coercivity of the multi-well potential.
In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.
The purpose of this paper is to prove an embedding theorem for Sobolev type functions whose gradients are in a Lorentz space, in the framework of abstract metric-measure spaces. We then apply this theorem to prove absolute continuity and differentiability of such functions.