Abstract
The Hausdorff measure with fractional index is used in order to define a functional on measurable sets of the plane. A fractal set, constructed using the well-known Von Koch set, is involved in the definition. This functional is proved to arise as the limit of a sequence of classical functionals defined on sets of finite perimeter. Thus it is shown that a natural extension of the ordinary functionals of the calculus of variations leads both to fractal sets and to the fractional Hausdorff measure.
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AMBROSIO, L., BRAIDES, A.: Functionals defined on partitions of sets of finite perimeter, I: integral representation and Γ-convergence. J. Math. Pures Appl.69 (3), 285–306 (1990)
AMBROSIO, L., BRAIDES, A.: Functionals defined on partitions of sets of finite perimeter, II: semicontinuity, relaxation and homogenization. J. Math. Pures Appl.69 (3), 307–333 (1990)
DE GIORGI, E., COLOMBINI, F., PICCININI, L.C.: Frontiere orientate di misura minima e questioni collegate. Quaderno della Scuola Normale Superiore, Pisa 1970
DE GIORGI, E., FRANZONI, T.: Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia3, 63–101 (1979)
FALCONER, K.J.: The geometry of fractals. Cambridge: Cambridge University Press 1985
FEDERER, H.: Geometric measure theory. New York: Springer 1969
MANDELBROT, B.B.: Les Objects fractals: forme, hasard et dimension. Paris: Flammarion 1975
GIAQUINTA, M., MODICA, G., SOUČEK, J.: Cartesian Currents, Weak Diffeomorphisms and Existence Theorems in Nonlinear Elasticity. Arch. Rat. Mech. Anal.106, 97–160 (1989)
GIUSTI E.: Minimal surfaces and function of bounded variation. Boston: Birkhäuser 1984
ROGERS, C.A.: Hausdorff measures. Cambridge: Cambridge University Press 1970
VISINTIN, A.: Generalized coarea formula and fractal sets. Japan J. Appl. Math., to appear
VISINTIN, A.: Models of pattern formation. C.R. Acad. Sci. Paris309 (I), 429–434 (1989)
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Braides, A., D’Ancona, P. Perimeter on fractal sets. Manuscripta Math 72, 5–25 (1991). https://doi.org/10.1007/BF02568263
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DOI: https://doi.org/10.1007/BF02568263