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Analysis and Numerics of Some Fractal Boundary Value Problems

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Analysis and Numerics of Partial Differential Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 4))

Abstract

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, 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.

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Acknowledgements

This work was supported by the NSF grant No. 1109356.

The author wishes to thank the authors of the papers presented in this note for their kindness in providing him with the preprints of their work.

The author wishes also to thank the Colleagues in Pavia who organized the Conference for their invitation and hospitality and the Editors of the Volume of the conference for the opportunity given to him of contributing this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA, 01609-2280, USA

    Umberto Mosco

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  1. Umberto Mosco
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Correspondence to Umberto Mosco .

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Editors and Affiliations

  1. Ist. Matematica Applicata e, Tecnologie Informatiche (IMATI), CNR Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Franco Brezzi

  2. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Piero Colli Franzone

  3. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Ugo Gianazza

  4. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Gianni Gilardi

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In memory of Enrico Magenes.

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Mosco, U. (2013). Analysis and Numerics of Some Fractal Boundary Value Problems. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_14

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