Geodesic
Analysis and Numerics of Some Fractal Boundary Value Problems
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 53-73.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We describe some recent results for boundary value problems with fractal boundaries. Our aim is to show that the numerical approach to boundary value problems, so much cherished and in many ways pioneering developed by Enrico Magenes, takes on a special relevance in the theory of boundary value problems in fractal domains and with fractal operators. In this theory, in fact, the discrete numerical analysis of the problem precedes the, and indeed give rise to, the asymptotic continuous problem, reverting in a sense the process consisting in deriving discrete approximations from the PDE itself by finite differences or finite elements. As an illustration of this point, in this note we describe some recent results on: the approximation of a fractal Laplacian by singular elliptic partial differential operators, by Vivaldi and the author; the asymptotic of degenerate Laplace equations in domains with a fractal boundary, by Capitanelli-Vivaldi; the fast heat conduction on a Koch interface, by Lancia-Vernole and co-authors. We point out that this paper has an illustrative purpose only and does not aim at providing a survey on the subject. 
@article{BUMI_2013_9_6_1_a2,
     author = {Mosco, Umberto},
     title = {Analysis and {Numerics} of {Some} {Fractal} {Boundary} {Value} {Problems}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {53--73},
     publisher = {mathdoc},
     volume = {Ser. 9, 6},
     number = {1},
     year = {2013},
     zbl = {1280.35035},
     mrnumber = {3077113},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a2/}
}
TY  - JOUR
AU  - Mosco, Umberto
TI  - Analysis and Numerics of Some Fractal Boundary Value Problems
JO  - Bollettino della Unione matematica italiana
PY  - 2013
SP  - 53
EP  - 73
VL  - 6
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a2/
LA  - en
ID  - BUMI_2013_9_6_1_a2
ER  - 
%0 Journal Article
%A Mosco, Umberto
%T Analysis and Numerics of Some Fractal Boundary Value Problems
%J Bollettino della Unione matematica italiana
%D 2013
%P 53-73
%V 6
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a2/
%G en
%F BUMI_2013_9_6_1_a2
Mosco, Umberto. Analysis and Numerics of Some Fractal Boundary Value Problems. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 53-73. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_1_a2/

[1] Y. Achdou - C. Sabot - N. Tchou, A multiscale numerical method for Poisson problems in some ramified domains with fractal boundary, Multiscale Model. Simul. 5 (3) (2006), 828-860. | DOI | MR | Zbl

[2] H. Attouch, Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program, London 1984. | MR | Zbl

[3] P.BAGNERINI - A. Buffa - E. Vacca, Finite elements for a prefractal transmission problem, C.R.A.S. Math, 342 (3) (2006), 211-214. | DOI | MR | Zbl

[4] M. T. Barlow - B. M. Hambly, Transition density estimates for Brownian motion on scale irregular Sierpiński gaskets, Ann. Inst. Henri Poincaré, 33 (5) (1997), 531-557. | fulltext EuDML | DOI | MR | Zbl

[5] J. R. Cannon - G. H. Meyer, On a diffusion in a fractured medium, SIAM J. Appl. Math., 3 (1971), 434-448. | Zbl

[6] R. Capitanelli - M. A. Vivaldi, Insulating layers and Robin problems on Koch mixtures, J. Diff. Eqn., 251 (4-5) (2011), 1332-1353. | DOI | MR | Zbl

[7] R. Capitanelli - M. A. Vivaldi, On the Laplacean transfer across a fractal mixture, Asymptotic Analysis, to appear. DOI 10.3233/ASY-2012-1149. | MR

[8] M. Cefalo - G. Della'Acqua - M.R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, Appl. Math. Comp., 218 (2012), 5453-5473. | DOI | MR | Zbl

[9] M. Cefalo - M.R. Lancia - H. Liang, Heat flow problems across fractal mixtures: regularity results of the solutions and numerical approximation, Differential and Integral Equations, in print. | MR | Zbl

[10] E. Evans, Extension Operators and Finite Elements for Fractal Boundary value Problems, PhD thesis, Worcester Polytecnic Institute, USA, 2011.

[11] P. Grisvard, Elliptic problems in nonsmooth domains, Advanced Publishing Program, London 1985. | MR | Zbl

[12] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. | DOI | MR | Zbl

[13] A. Jonsson, Dirichlet forms and Brownian motion penetrating fractals, Potential Analysis, 13 (2000), 69-80. | DOI | MR | Zbl

[14] A. Jonsson - H. Wallin, Function Spaces on Subsets of $\mathbb{R}^n$, Part 1 (Math. Reports) Vol. 2. London: Harwood Acad. Publ. 1984. | MR | Zbl

[15] K. Kuwae - T. Shioya, Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry, Comm. in Anal. and Geom., 11 (4) (2003), 599-673. | DOI | MR | Zbl

[16] M. R. Lancia, A transmission problem with a fractal interface, AZ. Anal. und Ihre Anwend., 21 (2002), 113-133. | DOI | MR | Zbl

[17] M. R. Lancia - E. Vacca, Numerical approximation of heat flow problems across highly conductive layers, ``Quaderni di Matematica'', Dept. Me. Mo. Mat., U. Roma La Sapienza, 2007. | MR

[18] M. R. Lancia - M. A. Vivaldi, Asymptotic convergence for energy forms, Adv. Math. Sc. Appl., 13 (2003), 315-341. | MR | Zbl

[19] M. R. Lancia - P. Vernole, Convergence results for parabolic transmission problems across highly conductive layers with small capacity, AMSA, 16 (2006), 411-445. | MR | Zbl

[20] M. R. Lancia - P. Vernole, Irregular heat flow problems, SIAM J. Math. Anal., 42 (4), (2010), 1539-1567. | DOI | MR | Zbl

[21] M. R. Lancia - P. Vernole, Semilinear evolution transmission problems across fractal layers, Nonlinear Anal. Th. and Appl., 75 (2012), 4222-4240. | DOI | MR | Zbl

[22] M. R. Lancia - P. Vernole, Semilinear fractal problems: approximation and regularity results, Nonlinear Anal. Th. and Appl., to appear. DOI: 10.1016/j.na.2012.08.020. | DOI | MR

[23] H. Liang, On the construction of certain fractal mixtures, MS Thesis, WPI, 2009.

[24] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123, 2 (1994), 368-421. | DOI | MR | Zbl

[25] U. Mosco, Variational fractals, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) Vol. XXV (1997), 683-712. | fulltext EuDML | MR

[26] U. Mosco, Harnack inequalities on scale irregular Sierpiński gaskets, Nonlinear Problems in Math. Physics and Related Topics II, Edited by Birman et al., Kluwer Acad./Plenum Publ. (New York, 2002), 305-328. | DOI | MR | Zbl

[27] U.MOSCO - M. A. Vivaldi, An example of fractal singular homogenization, Georgian Math. J. 14 (1) (2007), 169-194. | MR | Zbl

[28] U. Mosco - M. A. Vivaldi, Fractal Reinforcement of Elastic Membranes, Arch. Rational Mech. Anal., 194 (2009), 49-74. | DOI | MR | Zbl

[29] U. Mosco - M. A. Vivaldi, Vanishing viscosity for fractal sets, Discrete and Continuous Dynamical Systems. 28 (3) (2011), 1207-1235. | DOI | MR | Zbl

[30] U. Mosco - M. A. Vivaldi, Thin fractal fibers, Math. Methods in the Applied Sciences, to appear. DOI: 10.1002/mma.1621. | MR

[31] H. Pham Huy - E. Sanchez Palencia, Phénomènes des transmission à travers des couches minces de conductivité élevée, J. Math Anal. Appl., 47 (1974), 284-309. | DOI | MR | Zbl

[32] E. Vacca, Galerkin Approximation for Highly Conductive Layers, PhD Thesis, Dept. Me. Mo. Mat., U. Roma La Sapienza, 2005.

[33] R. Wasyk, Numerical solution of a transmission problem with pre fractal interface, PhD Thesis, WPI, 2007.