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Some results on surface measures in calculus of variations. (English) Zbl 0890.49020
Starting from the identity, for every smooth bounded open subset \(E\) of \(\mathbb{R}^n\), of the notions of perimeter, \((n-1)\)-dimensional Hausdorff measure, Minkowski content and the approximability in the sense of \(\Gamma\)-convergence of the perimeter by a sequence of elliptic functionals with a double equal well potential term, in the paper a similar study is carried out when \(\mathbb{R}^n\) is endowed with a convex Finsler metric depending continuously on the position.
Given such a metric \(\phi\) and introduced the integrated distance \(\delta_\phi\) associated to \(\phi\) and the \((n-1)\)-dimensional Hausdorff measure \(H^{n- 1}_\phi\) with respect to \(\delta_\phi\), a representation formula for \(H^{n-1}_\phi\) as an integral in the measure \(H^{n-1}\) is first proved. Then the perimeter \(P_\phi(E)\) of a set \(E\) with respect to \(\phi\) is introduced and some inequalities between \(P_\phi\) and \(H^{n-1}_\phi\) are proved. It is also shown that \(H^{n- 1}_\phi\) is a perimeter with respect to a suitable metric \(\psi\) constructed starting from \(\phi\). Then, for every set \(E\), \(P_\phi(E)\) is compared with the Minkowski content \(M^-_\phi(\partial E)\) with respect to \(\phi\) and it is proved that \(P_\phi(E)\leq M^-_\phi(\partial E)\) and that equality holds under suitable regularity assumptions.
Finally, the approximability of \(P_\phi(E)\) in the sense of \(\Gamma\)-convergence by a sequence of elliptic functionals with a double equal well potential term is discussed and the notions of normal and of mean curvature of a smooth hypersurface with respect to \(\phi\) are introduced.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
28A78 Hausdorff and packing measures
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