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On the total variation of the Jacobian. (English) Zbl 1041.49016
Let \(\Omega\) be an open subset of \({\mathbb R}^2\), and \(u=(u^1,u^2)\in L^\infty_{\text{ loc}}(\Omega;{\mathbb R}^2)\cap W^{1,p}(\Omega;{\mathbb R}^2)\) for some \(p>1\). In the paper, a comparison is carried out among the classical Jacobian determinant \(\det Du\) defined a.e. in \(\Omega\), the distributional Jacobian determinant \(\text{ Det} Du\) defined by \[ \langle \text{ Det} Du,\varphi\rangle= -\int_\Omega \{u^1D_2u^2D_1\varphi-u^1D_1u^2D_2\varphi\} \,dx,\quad\varphi\in C_0^\infty(\Omega), \] and the total variation \(TV(u,\Omega)\) of the Jacobian determinant given by \[ TV(u,\Omega)=\inf\left\{\liminf_{h\to+\infty} \int_\Omega| \det Du_h| dx : \{u_h\}\subseteq W^{1,2}(\Omega; {\mathbb R}^2),u_h\to u\text{ weakly in }W^{1,p}(\Omega;{\mathbb R}^2)\right\}. \] An explicit characterization of \(TV(u,\Omega)\) is given under additional assumptions on \(u\), and it is proved that \(TV(u,\Omega)\) can be expressed by means of the topological degree, provided \(u\) is also locally Lipschitz outside a finite number of points of \(\Omega\).
Some examples are also discussed proving nonidentity among the three notions of Jacobian determinant.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
74B20 Nonlinear elasticity
74G70 Stress concentrations, singularities in solid mechanics
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