We prove partial regularity for minimizers to elasticity type energies with p-growth, p > 1, in a geometrically linear framework in dimension n >= 3. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than n-(p * boolean AND 2), and actually n-2 in the autonomous case (full regularity is well-known in dimension 2). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions 2 and 3.
A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems
Iurlano F
2019-01-01
Abstract
We prove partial regularity for minimizers to elasticity type energies with p-growth, p > 1, in a geometrically linear framework in dimension n >= 3. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than n-(p * boolean AND 2), and actually n-2 in the autonomous case (full regularity is well-known in dimension 2). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions 2 and 3.
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