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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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References:
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