×

Nahm’s equations and free-boundary problems. (English) Zbl 1211.58015

García-Prada, Oscar (ed.) et al., The many facets of geometry. A tribute to Nigel Hitchin. Oxford: Oxford University Press (ISBN 978-0-19-953492-0/hbk). 71-91 (2010).
The basic motivation for this paper is the question of formulating an analogous theory for the Nahm equations associated to the infinite-dimensional Lie group of area-preserving diffeomorphisms of a surface.
Let \(X\) be a compact-oriented Riemannian manifold. The main purpose in this work is to ask the following
Question: Is there a pair \(H_0\leq H_1\) and a function \(\theta\) on the set \(\Omega_{H_0,H_1}\subset X\times{\mathbb R}\) with \(\theta=0,1\) on the hypersurfaces \(\{z=H_0\}\), \(\{z=H_1\}\), with fluxes \(\rho_i\) and \(\Delta_\varepsilon\theta =0\)? If so, is the solution essentially unique?
Three equivalent problems of the above mentioned question associated to a compact Riemannian manifold are also formulated. In the final part of the present paper, some remarks about existence results and a comparison with the free-boundary literature, as well as the relationship with Nahm’s equations are provided.
For the entire collection see [Zbl 1192.00076].

MSC:

58J32 Boundary value problems on manifolds
35R35 Free boundary problems for PDEs
PDFBibTeX XMLCite
Full Text: arXiv