A Carleman function and the Cauchy problem for the Laplace equation.
A conforming finite element method with Lagrange multipliers for the biharmonic problem
A free boundary problem related to single-layer potentials.
A free boundary value problem in potential theory
This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain and a function defined in satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of . The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists a unique solution...
A heat approximation
The Fourier problem on planar domains with time variable boundary is considered using integral equations. A simple numerical method for the integral equation is described and the convergence of the method is proved. It is shown how to approximate the solution of the Fourier problem and how to estimate the error. A numerical example is given.
A note on the parabolic variation
A condition for solvability of an integral equation which is connected with the first boundary value problem for the heat equation is investigated. It is shown that if this condition is fulfilled then the boundary considered is -Holder. Further, some simple concrete examples are examined.
A numerical solution of the Dirichlet problem on some special doubly connected regions
The aim of this paper is to give a convergence proof of a numerical method for the Dirichlet problem on doubly connected plane regions using the method of reflection across the exterior boundary curve (which is analytic) combined with integral equations extended over the interior boundary curve (which may be irregular with infinitely many angular points).
A Nyström method for boundary integral equations in domains with corners.
A remark on uniqueness of best rational approximants of degree 1 in of the circle.
A symmetry problem
Consider the Newtonian potential of a homogeneous bounded body D ⊂ ℝ³ with known constant density and connected complement. If this potential equals c/|x| in a neighborhood of infinity, where c>0 is a constant, then the body is a ball. This known result is now proved by a different simple method. The method can be applied to other problems.