Analysis and Numerics of Some Fractal Boundary Value Problems

Chapter
Part of the Springer INdAM Series book 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.

Keywords

Spectral Measure Dirichlet Form Dirichlet Condition Transmission Problem Trace Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

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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Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

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