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Periodic homogenisation of Hamilton–Jacobi equations: 2. Eikonal equations

Published online by Cambridge University Press:  14 November 2011

Marie C. Concordel
Marie C. Concordel
Affiliation:
M.S.R.I., 1000 Centennial Drive, Berkeley, CA 94720, U.S.A e-mail: [email protected]
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Abstract

Homogenisation of the first-order Hamilton-Jacobi equations when H is periodic in the second variable, leads to an effective Hamiltonian H satisfying: uε converges, as ε → 0, to the solution u of ut + H(Du) = 0. In our first paper, we assumed that H is convex and we derived a variational formula giving H. In this second paper, we consider eikonal equations, i.e. H(p, x) = ½|p|2 – V(x). Using our variational formula, we compute explicitly the effective Hamiltonian in several cases, and we study precisely the lack of strict convexity for H (‘flat part’ around the origin).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 665 - 689
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

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