Curvature penalization of strongly anisotropic interfaces models and their phase-field approximation

TL;DR

The paper tackles the lack of lower semicontinuity in strongly anisotropic interfacial energies by adding a curvature term and by developing phase-field approximations that recover the sharp-interface limits. It proves lower semicontinuity and Γ-convergence results for anisotropic perimeters with curvature and for anisotropic Mumford–Shah functionals, extending the analysis to free-discontinuity structures such as crack tips and junctions through a varifold-based framework. Central to the results is a phase-field model that combines an anisotropic Modica–Mortola density with Willmore-type curvature, together with a secondary phase field to encode endpoint energies, and a generalized Gauss–Bonnet analysis for varifolds. The work advances the mathematical understanding of curvature-penalized, strongly anisotropic interfaces and provides rigorous convergence results that connect diffuse-interface models to sharp-interface energetics with singular crack geometries.

Abstract

This paper studies the effect of anisotropy on sharp or diffuse interfaces models. When the surface tension is a convex function of the normal to the interface, the anisotropy is said to be weak. This usually ensures the lower semicontinuity of the associated energy. If, however, the surface tension depends on the normal in a nonconvex way, this so-called strong anisotropy may lead to instabilities related to the lack of lower semicontinuity of the functional. We investigate the regularizing effects of adding a higher order term of Willmore type to the energy. We consider two types of problems. The first one is an anisotropic nonconvex generalization of the perimeter, and the second one is an anisotropic nonconvex Mumford-Shah functional. In both cases, lower semicontinuity properties of the energies with respect to a natural mode of convergence are established, as well as $Γ$-convergence type results by means of a phase field approximation. In comparison with related results for curvature dependent energies, one of the original aspects of our work is that, in the context of free discontinuity problems, we are able to consider singular structures such as crack-tips or multiple junctions.

Figures (6)

• Figure 1: The triple junction configuration
• Figure 2: Blow up at $x_0 \in \mathcal{P}_\Gamma$.
• Figure 3: Definition of $\overline{v}_\epsilon$ (left) and $\overline{w}_\epsilon$ (right) according to offsets of $J_u$ and $\mathcal{P}_{J_u}$.
• Figure 4: Profile of $\overline{v}_\epsilon$ (left) and $\overline{w}_\epsilon$ (right) accross the respective dotted lines in Figure \ref{['figOffsetsRecovery']}.
• Figure 5:

Theorems & Definitions (55)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5