Calculus of Variations and Geometric Measure Theory

T. Esposito

Second-order approximation of free-discontinuity problems with linear growth

created by esposito1 on 09 Sep 2016


Submitted Paper

Inserted: 9 sep 2016
Last Updated: 9 sep 2016

Year: 2016


Motivated by applications to image denoising, we propose an approximation of functionals of the form $ F(u)=\int_\Omega \mid\nabla u\mid\,dx+\int_{S_u}g(\mid u^+-u^-\mid)\,d\mathcal{H}^{n-1}+\mid D^cu\mid(\Omega),\; u\in BV(\Omega), $ with $g\colon [0,+\infty)\to[0,+\infty)$ increasing and bounded. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian or on the Laplacian of the edge variable $v$ which thus belongs to $W^{2,2}(\Omega)$. When the space dimension is equal to two and three $v$ is then continuous and this improved regularity leads to a sequence of approximating functionals which are ready to be used for numerical simulations.

Keywords: Free-discontinuity problems, Ambrosio-Tortorelli approximation, $\Gamma\hbox{-}$convergence