1,356 research outputs found
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications
We propose a new variational model in weighted Sobolev spaces with
non-standard weights and applications to image processing. We show that these
weights are, in general, not of Muckenhoupt type and therefore the classical
analysis tools may not apply. For special cases of the weights, the resulting
variational problem is known to be equivalent to the fractional Poisson
problem. The trace space for the weighted Sobolev space is identified to be
embedded in a weighted space. We propose a finite element scheme to solve
the Euler-Lagrange equations, and for the image denoising application we
propose an algorithm to identify the unknown weights. The approach is
illustrated on several test problems and it yields better results when compared
to the existing total variation techniques
Interior penalty discontinuous Galerkin FEM for the -Laplacian
In this paper we construct an "Interior Penalty" Discontinuous Galerkin
method to approximate the minimizer of a variational problem related to the
Laplacian. The function is log H\"{o}lder
continuous and . We prove that the minimizers of the
discrete functional converge to the solution. We also make some numerical
experiments in dimension one to compare this method with the Conforming
Galerkin Method, in the case where is close to one. This example is
motivated by its applications to image processing.Comment: 26 pages, 2 figure
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