Corner singularities between free surfaces and open boundaries

We consider the solution of the Stokes problem at a corner between a free surface and an inflow or outflow boundary. A formal asymptotic solution for the dominant contribution to the streamfunction near the corner is derived. We give a heuristic discussion of the relevance of the nature of the corner singularity to the formulation of well-posed boundary value problems.     

Geometric properties of minimal surfaces with free boundaries

Mathematische Zeitschrift -     

On stable surfaces of prescribed mean curvature with partially free boundaries

We study surfaces of prescribed bounded mean curvature H in a partially free boundary configuration . If is projectable onto and is a cylinder surface over the x¹,x²-plane, we show that also the spanned H-surface is projectable onto this plane. Besides certain conditions on and H, we have to suppose that is stationary and freely stable w.r.t. the generalized area functional , and the vector field is assumed to be tangential on . Consequences are uniqueness of freely stable H-surfaces and solvability of a mixed boundary problem for the nonparametric prescribed mean curvature equation. Mathematics Subject Classification (2000): 53 A 10, 35 J 65, 35 R 35, 49 Q 05     

Asymptotic analysis for surfaces with large constant mean curvature and free boundaries

We prove that simply connected H-surfaces with bounded area and free boundary in a domain necessarily concentrate at a critical point of the mean curvature of the boundary of this domain.     

On the regularity of minimal surfaces with free boundaries in Riemannian manifolds

We show c^1,-regularity of minimal surfaces in Riemannian manifolds with a free boundary on C²-hypersurfaces with bounded second fundamental form and a uniform neighborhood on which the nearest point projection is uniquely defined and differentiable. The decisive step is the proof of continuity at the free boundary.partially supported by SFB 72 (Deutsche Forschungsgemeinschaft)     

Approximation of time-dependent free boundaries

Summary Two problems are considered in the paper: the first of them is connected with elliptic variational inequalities and consists in developing a moving obstacle algorithm for approximating the unknown free boundary; the other problem is linked with numerical solution of the Stefan problem, which is formulated in the similar way as in the elliptic case. Some computational aspects are also discussed in the paper.     

Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993     

Gluings of surfaces with polygonal boundaries

By pairwise gluing edges of a polygon, one obtains two-dimensional surfaces with handles and holes. We compute the number N _g,L (n ₁, ..., n _L ) of distinct ways to obtain a surface of given genus g whose boundary consists of L polygonal components with given numbers n ₁, ..., n _L of edges. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursion relations between the N _g,L . We show that the Harer-Zagier numbers arise as a special case of N _g,L and derive a new closed-form expression for them.     

The homogenization of surfaces and boundaries

In this survey I would like to describe several homogenization issues that arise when considering surfaces, embedded in a periodic media, that satisfy some associated geometrical property like minimizing a “periodic area integral,” or that are an interphase of a flow, a flame, a drop, whose shape is influenced, again by the surrounding periodic media.     

One-dimensional problems with free boundaries in ecology

Statements of problems with free boundaries are given for nonlinear parabolic equations arising in ecology and medicine. Some constructive methods for their solution are considered. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 993–997, July, 1997.