Existence of minimizers for a class of anisotropic free discontinuity problems
The paper is concerned with the existence of minimizing pairs \((u,K)\) for the generalized Mumford--Shah functional \[ \int_{\Omega\setminus K}f(x,\nabla u)+\alpha(u-g)^2 dx +\beta{\mathcal H}^{n-1}(K\cap\Omega). \] under uniform continuity assumptions on \(f(\cdot,p)\). The idea of the proof is to estabilish a density lower bound \[ {\mathcal H}^{n-1}(S_u\cap B_\rho(x))\geq\theta\rho^{n-1} \qquad\forall x\in \overline{S}_u, B_\rho(x)\subset \Omega \] for minimizers \(u\) of the weak formulation of the problem in the space of special functions with bounded variation. The density lower bound implies that \({\mathcal H}^{n-1}(\Omega\cap\overline{S}_u\setminus S_u)=0\), and hence that \((u,\overline{S}_u)\) is a minimizing pair. The proof of the density lower bound, following the original approach of De Giorgi, Carriero and Leaci, is related to the rate of decay in \(\rho\) of the energy \[ \int_{B_\rho(x)}f(x,\nabla v(x)) dx \] for local minimizers \(v\) of \(v\mapsto\int f(x,\nabla v)\) in a Sobolev space. In this particular situation a recent regularity result of Cupini, Fusco and Petti leads to the decay estimate.
scientific article; zbMATH DE number 1472054