×

zbMATH — the first resource for mathematics

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.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] B. Chalmers, Principles of Solidification. Wiley, New York, 1964.
[2] E. De Giorgi, F. Colombinie, L.C. Piccinini, Frontiere Orientate di Misura Minima e Questioni Collegate. Editrice Tecnico-Scientifica, Pisa, 1972. · Zbl 0296.49031
[3] K.J. Falconer, The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1985. · Zbl 0587.28004
[4] H. Federer, Geometric Measure Theory. Springer-Verlag, Berlin, 1969. · Zbl 0176.00801
[5] M.C. Flemings, Solidification Processing. McGraw-Hill, New York, 1974.
[6] W.H. Fleming and R. Rishel, An integral formula for total gradient variation. Arch. Math.,11 (1960), 218–222. · Zbl 0094.26301 · doi:10.1007/BF01236935
[7] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston, 1984. · Zbl 0545.49018
[8] M.E. Gurtin, On a theory of phase transitions with interfacial energy. Arch. Rational Mech. Anal.,87 (1985), 187–212. · doi:10.1007/BF00250724
[9] B.B. Mandelbrot, Fractals. Form, Chance and Dimension. Freeman, San Francisco, 1977. · Zbl 0376.28020
[10] A. Visintin, Surface tension effects in phase transitions. Material Instabilities in Continuum Mechanics and Related Mathematical Problems (ed. J. Ball), Clarendon Press, Oxford, 1988, 505–537. · Zbl 0648.73050
[11] A. Visintin, Non-convex functionals related to multi-phase systems. S.I.A.M.T. Math. Anal.,21 (1990), 1281–1304. · Zbl 0723.49006 · doi:10.1137/0521071
[12] A. Visintin, Surface tension effects in two-phase systems. Proceedings of the Colloquium on Free Boundary Problems held in Irsee in June 1987 (forthcoming).
[13] A. Visintin, Pattern Evolution. Ann. Scuola Norm. Sup. Pisa,17 (1990), 197–225. · Zbl 0712.49039
[14] A. Visintin, Generalized coarea formula. Recent Advances in Nonlinear Elliptic and Parabolic Problems (eds. P. Benilan, M. Chipot, L.C. Evans and M. Pierre), Longman, Harlow, 1989. · Zbl 0701.49045
[15] A. Visintin, Models of pattern formation. C.R. Acad. Sci. Paris,309 (1989), Sér. I, 429–434. · Zbl 0749.49026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.