Variational Formulation of the Spatially Variant Fractional Laplacian and Optimal Parameter Selection


We provide a variational definition for the spatially variant fractional Laplacian and study the well-posedness of the associated elliptic equation. The analysis of the problem requires the use of non-standard Sobolev spaces with weights that are not of Muckenhoupt-type, and related boundary trace results. We further establish increased regularity results for the solution to the elliptic problem and consider the use of the fractional operator as a regularizer in mathematical imaging. The latter leads to the optimal selection of the fractional order in image reconstruction. We finalize the talk with numerical examples and future research directions. 


Carlos Rautenberg is Assistant Professor at George Mason University on the Department of Mathematical Sciences. His research interests involve several areas of analysis and applied mathematics: optimal control, variational and quasi-variational inequalities, optimal filtering, parameter estimation, sensor placement, Riccati equations and mathematical imaging.

Previously, he has been ECMath (Einstein Center for Mathematics Berlin) Junior Research Group Leader at the Department of Mathematics of the Humboldt Universität zu Berlin, and former deputy head of Research Group 8 at Weierstrass Institute for Applied Analysis and Stochastics in Berlin, Germany.


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