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

Article type: Research Article

Authors: Esposito, Teresa^{; *}

Affiliations: Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: Motivated by applications to image denoising, we propose an approximation of functionals of the form F(u)=∫Ω|∇u|dx+∫Sug(|u+−u−|)dHn−1+|Dcu|(Ω),u∈BV(Ω), with g:[0,+∞)→[0,+∞) 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 W2,2(Ω). 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: Ambrosio–Tortorelli approximation, free-discontinuity problems, Γ-convergence

DOI: 10.3233/ASY-181476

Journal: Asymptotic Analysis, vol. 110, no. 1-2, pp. 21-52, 2018