Regularity for minimizers of the Griffith fracture energy
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Abstract: Using the Griffith energy, fracture within a brittle elastic material can be modeled as an energy minimization problem. Coming from the Dark Side (\`a la Mingione), we are interested in the regularity of minimizers and, in particular, the structure of the crack. For this, we must unravel the complex interplay of bulk (elastic) and surface (crack) energies in the vectorial setting of linearized frame indifference. In dimension 2, we prove that the crack of a minimizer is given by a $C^{1,1/2}$ surface outside of a singular set of points with dimension strictly less than 1, analogous to results for the scalar-valued Mumford-Shah functional. Joint work with Manuel Friedrich (FAU) and Camille Labourie (UPS)