Sobolev spaces with non-Muckenhoupt weights, fractional elliptic\n operators, and applications

2018/03/27 by Harbir Antil, Antil, Harbir, Carlos N. Rautenberg +1
Engineering · Immunology and Microbiology · #Numerical methods in engineering #Biomarkers in Disease Mechanisms #Elasticity and Material Modeling

paper · pdf · doi:10.48550/arxiv.1803.10350

Abstract

We propose a new variational model in weighted Sobolev spaces with\nnon-standard weights and applications to image processing. We show that these\nweights are, in general, not of Muckenhoupt type and therefore the classical\nanalysis tools may not apply. For special cases of the weights, the resulting\nvariational problem is known to be equivalent to the fractional Poisson\nproblem. The trace space for the weighted Sobolev space is identified to be\nembedded in a weighted L² space. We propose a finite element scheme to solve\nthe Euler-Lagrange equations, and for the image denoising application we\npropose an algorithm to identify the unknown weights. The approach is\nillustrated on several test problems and it yields better results when compared\nto the existing total variation techniques.\n

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