Calculus of Variations and Geometric Measure Theory

M. Fornasier - R. March - F. Solombrino

Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data

created by solombrin on 30 May 2011
modified on 27 Sep 2013


Published Paper

Inserted: 30 may 2011
Last Updated: 27 sep 2013

Journal: Annali di Matematica Pura e Applicata
Volume: 192
Number: 3
Pages: 361-391
Year: 2013
Doi: 10.1007/s10231-011-0228-8


We consider the regularization of linear inverse problems in fracture mechanics and image processing by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. The discrepancy term penalizes the $L^2$ distance between a datum and a version of the unknown function which is filtered by means of a non-invertible linear operator. Depending on the type of the involved operator the resulting variational problem has had several applications: image {\it deblurring}, or {\it inverse source problems} in the case of convolution operators, and image {\it inpainting} in the case of suitable local operators, as well as the modelling of {\it propagation of fracture}. We present counterexamples showing that, despite this regularization, the problem is actually in general ill-posed. We provide however existence results of minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets in two dimensions.

The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces.