arXiv subjects

Djamel eddine Kebiche

Publications and source records attributed to Djamel eddine Kebiche.

3 recordsLinked to original sources

arXiv2025

A new approach to weighted Sobolev spaces

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré Inequality and a trace operator are developed, and density results of smooth function are established.

math.AP↗

arXiv2024

The Hahn-Banach theorem in spaces of nonlinear generalized functions

In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined $\varepsilon$-wise, which simplifies the framework and makes the extension of linear functionals more manageable. As an application of our main result, we demonstrate the separation of convex sets in Colombeau spaces.

math.FA↗

arXiv2024

Universal properties of spaces of generalized functions

By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal property of Schwartz distributions, as the simplest way to have derivatives of continuous functions, Colombeau algebra as the simplest quotient algebra where representatives of zero are infinitesimal, and generalized smooth functions as the universal way to associate set-theoretical maps of non-Archimedean numbers defined by nets of smooth functions (e.g. regularizations of distributions) and having arbitrary derivatives. Each one of these properties yields a characterization up to isomorphisms of the corresponding space. The paper requires only the notions of category, functor, natural transformation and Schwartz's distributions, and introduces the notion of universal solution using a simple and non-abstract language.

math.FA↗