1,356 research outputs found

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

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    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 L2L^2 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 p(x)p(x)-Laplacian

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    In this paper we construct an "Interior Penalty" Discontinuous Galerkin method to approximate the minimizer of a variational problem related to the p(x)p(x)-Laplacian. The function p:Ω[p1,p2]p:\Omega\to [p_1,p_2] is log H\"{o}lder continuous and 1<p1p2<1<p_1\leq p_2<\infty. 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 p1p_1 is close to one. This example is motivated by its applications to image processing.Comment: 26 pages, 2 figure
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