Formation of cracks under deformations with finite energy

Abstract.

With a map $f: \Omega\to {\bf R}^n$, $\Omega\subset {\bf R}^n$, that belongs to the John Ball class $A_{p,q}^{ + }(\Omega)$ where n-1 < p < n and $q\geq p/(p-1)$ one can associate a set valued map F whose values $F(x)\subset {\bf R}^n$ are subsets of ${\bf R}^n$ describing the topological character of the singularity of f at $x\in\Omega$. Šverak conjectured that ${\cal H}^{n-1}(F(S)) = 0$, where S is the set of points at which f is not continuous and ${\cal H}^{n-1}$ is the Hausdorff measure. The purpose of our paper is to confirm this expectation.