Análisis de un problema de optimización en espacios de exponente variable con aplicaciones en el filtrado de ruido en imágenes

Abstract

In this work, we study the main properties of the Lebesgue and Sobolev spaces with a variable exponent. Further, based on this study we propose an optimization problema associated to image denoising.

In order to study existence and uniqueness of solutions for this problem, which involves a functional with variable exponent, we use the direct method in the calculus of variations. In this aim, a brief review of functional analysis, measure theory, Lesbesgue and Sobolev spaces, and calculus of variations is carried out. In particular, we focus on the study of semi-modular spaces and amalgam spaces, which are essential tools for construction and treatment of this application.