Report: EPSRC Visiting Fellowship GR/N23288

The aim of this short visit to the UK by Professor Obnosov was to identify overlap between the mathematical approaches underlying Fuchsian differential equations applied to free boundary problems (Craster 1997) and Riemann- Hilbert problems applied to doubly periodic structures in effective medium theory (Obnosov 1999). The double periodicity has, in the past, rendered exact solutions to model problems very difficult to find. It was hoped that the overlapping expertise would bring new techniques and ideas to each subject. It turns out that there is indeed some overlap, but that it is much more advantageous to tackle the doubly periodic problems using a novel approach that was developed (Craster 2000) with this visit in mind. During this visit we mainly concentrated upon this approach and double periodicity.

We have developed a simple method capable of tackling any rectangular doubly-periodic checkerboard media and this allows us to find explicit and neat formulae providing benchmark solutions. The novel idea is to use a mapping involving a two-sheeted Riemann surface. This has now led to a joint publication, Craster & Obnosov (2000a), within which we explicitly find closed-form formulae for effective resistivities of four phase checkerboard media, and as a by-product prove a conjecture by Mortola & Steffé (1985). During this visit we revised the Craster & Obnosov paper, and pursued the analysis further by modelling debonding between the phases, or perfect transmission, once again finding neat closed-form solutions, Craster & Obnosov (2000b). Several other problems were explored in various levels of detail, but have yet to come to fruition.

During his visit Professor Obnosov gave seminars in the Mathematics department at Imperial College and at OCIAM in Oxford; both talks appear to have been well received. Within the two month visit we had originally hoped to just focus upon good collaborative projects for the future, fortunately one of these has worked nicely and some useful papers are emerging.

References

R. V. Craster, The solution of a class of free boundary problems, Proc. Roy. Soc. Lond. A, 453, (1997) , pp. 607-630

R. V. Craster, On effective parameters for periodic checkerboard composites, Proc. Roy. Soc. Lond. A, 456 (2000), pp. 2741-2754.

R. V. Craster and Y. V. Obnosov, Four phase checkerboard composites. Accepted subject to revision SIAM J. Appl. Maths., 2000a.

R. V. Craster and Y. V. Obnosov, Checkerboard composites with separated phases. Submitted to J. Math. Phys., 2000b

S. Mortola and S. Steffé, A two-dimensional homogenization problem, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali. Serie VIII 78, 77-82 (1985).

Y. V. Obnosov, Periodic heterogeneous structures: new explicit solutions and effective characeristics of refraction of an imposed field, SIAM J. Appl. Math., 59 (1999), 1267-1287


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