Geometry and combinatorics of PS-3 integral equations

More than a hundred years ago H.Poincare and V.A.Steklov considered a problem for the Laplace equation with spectral parameter in the boundary conditions. Today similar problems for two adjacent domains with the spectral parameter in the conditions on the common boundary of the domains arises in a variety of situations: in justification and optimization of domain decomposition method, simple 2D models of oil extraction. Singular 1D integral Poincare-Steklov equation naturally emerges after reducing this 2D problem to the common boundary of the domains.

This latter equation has one functional parameter R(x) - the smooth change of variable on the interval of integration. When R(x) is a rational function, the considered equation is equivalent to some Riemann monodromy problem [1], so powerful geometrical techniques may be applied for the investigation. Explicit representations for the eigenvalues/eigenfunctions may be given: in terms of Jacobi elliptic functions when deg R=2 [2] or in terms of Kleinian membranes when deg R=3 [3].

In the latter case the Riemann monodromy problem may be reformulated in terms of branched projective structures on genus 2 Riemann surface. The problem has essentially combinatorial nature and may be solved with the help of the technique resembling Grothendieck's dessins.

[1] Bogatyrev A.B. Poincare-Steklov integral equations and Riemann monodromy problem, Functional Analysis and Applications, 34:2 (2000), pp. 9-22.
[2] Bogatyrev A.B. A geometric method for solving a series of integral PS equations, Math. Notes, 63:3 (1998), pp. 302-310.
[3] Bogatyrev A.B. PS-3 integral equations and projective structures on Riemann Surfaces, Math. Sbornik, 192:4 (2001), pp. 3--36.