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Generalized coarea formula and fractal sets. (English) Zbl 0736.49030
The author introduces a class of functionals \(\Lambda: L^1(\Omega)\to[0,+\infty]\), \(\Omega\subset\mathbb{R}^n\), for which the following “generalized coarea” formula holds:
\[ \Lambda(u) = \int_{\mathbb{R}}\Lambda(H_s(u))\,ds, \quad u\in L^1(\Omega); \]
here \(H_s(t)=0\) if \(t<s\) and \(H_s(t)=1\) if \(t\geq s\), \(t,s\in\mathbb{R}\). The author studies interesting properties of this class and gives some applications to models of surface tension effects in two-phase systems. A typical example is given by the generalized variation \(V(u)=\int_ \Omega| \nabla u|\), which is equivalent to the Cesari variation of \(u\) [L. Cesari, Ann. Scuola Norm. Super. Pisa 5, 299–313 (1936; Zbl 0014.29605)]. It is well-known that if \(A\) is a measurable set, \(V(\chi_A)\) represents the generalized perimeter of \(A\) [see the reviewer, Ann. Scuola Norm. Super. Pisa, Cl. Sci. Fis. Mat., III. Ser. 18, 201–231 (1964; Zbl 0129.10705)] and so the functional \(V\) can be used to represent the surface tension contribution to the free enthalpy of these systems. Moreover two definitions of fractional dimension for set boundaries are given and the relations with other classical definitions are explained.

49Q20 Variational problems in a geometric measure-theoretic setting
Full Text: DOI
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