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Topological degree, Jacobian determinants and relaxation. (English) Zbl 1177.49066
Summary: A characterization of the total variation $$TV (u,\Omega)$$ of the Jacobian determinant $$\det Du$$ is obtained for some classes of functions $$u: \Omega \to \mathbb{R}^n$$ outside the traditional regularity space $$W^{1,n} (\Omega; \mathbb{R}^{n})$$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity $$x_{0} \in \Omega$$. Relations between $$TV(u, \Omega)$$ and the distributional determinant $$\text{Det} Du$$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $$u \in W^{1,p} (\Omega; \mathbb{R}^{n}) \cap W^{1,\infty} (\Omega \setminus \{x_{0}\}; \mathbb{R}^{n})$$.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 35J50 Variational methods for elliptic systems 49J35 Existence of solutions for minimax problems
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