[Paper Review] Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy

Antonin Chambolle, Sergio Conti+1 more|arXiv (Cornell University)|Oct 5, 2017

Advanced Mathematical Modeling in EngineeringComputer Science30 references32 citations

TL;DR

This paper establishes a dimensionally general approximation result for functions in GSBD with small jump sets, proving they can be closely approximated by smooth functions in a slightly smaller domain. This enables the first proof of existence of strong minimizers for Griffith's energy in linearized elasticity across all dimensions, extending prior 2D results by closing the jump set of local minimizers.

ABSTRACT

We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy.

Motivation & Objective

• To establish a general approximation result for functions in GSBD with small jump sets in arbitrary dimensions.
• To extend the existence of strong minimizers for Griffith's energy beyond two dimensions.
• To close the jump set of local minimizers in the GSBD framework, ensuring they are truly discontinuous on closed (n−1)-dimensional sets.
• To provide a foundation for the study of quasi-static fracture evolution and integral representations in SBD.

Proposed method

• A dyadic cube decomposition of the domain is used, classifying cubes as 'good' or 'bad' based on jump set measure.
• In good cubes, the function is approximated by an affine function outside a small exceptional set using a Poincaré-Korn-type inequality.
• A mollification procedure regularizes the function in good cubes, ensuring smoothness.
• A partition of unity combines the smooth approximation in good cubes with the original function in bad cubes.
• The construction ensures the approximation is smooth in a large compact subset of the domain.
• The method is applied to prove convergence of quasi-minimizers with vanishing jump sets, leading to existence of strong minimizers.

Experimental results

Research questions

• RQ1 Can functions in GSBD with small jump sets be approximated by smooth functions in any dimension n ≥ 2?
• RQ2 Does the jump set of a local minimizer of Griffith's energy remain closed in higher dimensions?
• RQ3 Can the existence of strong minimizers for Griffith's energy be established in n-dimensional linearized elasticity without artificial constraints?
• RQ4 How does the approximation of functions with small jump sets enable the closure of the jump set for minimizers?
• RQ5 What is the role of the Poincaré-Korn inequality in the approximation of SBD functions with small jump sets?

Key findings

• The paper establishes a n-dimensional approximation result for GSBD functions with small jump sets, showing they can be approximated by smooth functions in a slightly smaller domain.
• The approximation method ensures the resulting function is smooth in a large compact subset of the domain, with controlled energy and jump set.
• The key technical advance is a new slicing-based approximation technique that avoids reliance on coarea formula or planar-specific constructions.
• The method proves the closedness of the jump set for local minimizers of Griffith's energy in any dimension n.
• The result implies the existence of strong minimizers for Griffith's energy in the class A2, where the discontinuity set is closed and the displacement is C1 outside it.
• The proof relies on a compactness and semicontinuity result for sequences with vanishing jump sets, which is established via the new approximation scheme.