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A notion of total variation depending on a metric with discontinuous coefficients. (English) Zbl 0842.49016
The authors consider the problem of defining a perimeter of a set when \(\mathbb{R}^n\) is endowed with a general distance. More precisely, if \(\Omega\) is a Lipschitz bounded open subset of \(\mathbb{R}^n\) and \(\phi: \Omega\times \mathbb{R}^n\to [0, +\infty[\) is a Borel function with \(\phi(x,\cdot)\) convex and one-homogeneous, and such that \(\phi(x, z)\leq C|z|\), for every \(u\in \text{BV}(\Omega)\) the total variation of \(u\) with respect to \(\phi\) is defined by \[ \int_\Omega |Du|_\phi= \sup\Biggl\{ \int_\Omega u\text{ div } \sigma dx\Biggr\}, \] where the supremum is taken over all vector fields \(\sigma\in L^\infty(\Omega; \mathbb{R}^n)\) with compact support in \(\Omega\) such that \(\text{div } \sigma\in L^n(\Omega)\) and \(\phi^0(x, \sigma(x))\leq 1\) on \(\Omega\), \(\phi^0\) being the dual function of \(\phi\).
It is proven that \(\int_\Omega |Du|_\phi\) can be represented in an integral form with respect to the measure \(|Du|\). Moreover, if \({\mathcal F}[\phi]: \text{BV}(\Omega)\to [0, +\infty]\) is the functional defined by \[ {\mathcal F}[\phi](u)= \begin{cases} \int_\Omega \phi(x, \nabla u(x))dx\quad & \text{if } u\in W^{1,1}(\Omega),\\ + \infty\quad & \text{otherwise},\end{cases} \] and if \(\overline{{\mathcal F}[\phi]}: \text{BV}(\Omega)\to [0, +\infty]\) denotes the relaxed functional of \({\mathcal F}[\phi]\) in the topology \(L^1(\Omega)\), we have \[ \int_\Omega |Du|_\phi= \overline{{\mathcal F}[\phi]}(u)\qquad \forall u\in \text{BV}(\Omega). \] Finally, setting for all \(u\in \text{BV}(\Omega)\) \(\nu^u(x)= {dDu\over d|Du|} (x)\) and \({\mathcal I}[\phi](u)= \int_\Omega \phi(x, \nu^u(x)) d|Du|\) it is shown that the relaxed functional \(\overline{{\mathcal I}[\phi]}\) of \({\mathcal I}[\phi]\) on \(\text{BV}(\Omega)\) coincides with \(\overline{{\mathcal F}[\phi]}\) provided \(\phi\) is upper semicontinuous. On the contrary, when \(\phi\) is only a Borel function, denoting by \({\mathcal N}(\Omega)\) the class of Lebesgue negligible subsets of \(\Omega\) and setting for every \(N\in {\mathcal N}(\Omega),\) \[ \phi_N(x, z)= \begin{cases} \phi(x, z)\quad & \text{if } x\in \Omega\backslash N,\\ C|z|\quad & \text{if } x\in N,\end{cases} \] then for all \(u\in \text{BV}(\Omega)\), \[ \int_\Omega |Du|_\phi= \sup_{N\in {\mathcal N}(\Omega)} \overline{{\mathcal I}[\phi_N]} (u). \]
Reviewer: G.Buttazzo (Pisa)

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
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