Seminar

The purpose of this informal introductory lecture is to observe that the study of classical Stokes-waves belong to a class of geometric problems in which zero Dirichlet conditions for a harmonic function combine with the requirement that the normal derivative is a non-constant function of position to determine the free boundary. When surface tension is included, the normal derivative is determined by the the position and a constant multiple of the boundary curvature; when the liquid is bounded by an elastic sheet the free boundary is determined by requiring that the normal derivative depends on position and nonlinearly on the curvature.

In a later lecture the problem for steady waves with prescribed distribution of vorticity (prescribed rearrangement class) will be discussed. That the vorticity is a function of the stream function is the Lagrange multiplier rule corresponding to this prescription. Thus the stream function satisfies a semilinear Poisson equation in which the nonlinearity is not prescribed a priori; it is part of the solution.