### A Carleman function and the Cauchy problem for the Laplace equation.

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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 $\Omega $ and a function $u$ defined in $\Omega $ satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of $\Omega $. 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...

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 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 $\frac{1}{2}$-Holder. Further, some simple concrete examples are examined.

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).

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.

A technique is developed for constructing the solution of ${\Delta}^{2}u=F$ in $R=\left\{\right(x,y):|x|\<a,\phantom{\rule{0.277778em}{0ex}}|y|\<b\}$, subject to boundary conditions $u=\phi $, $\frac{\partial u}{\partial n}=\psi $ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in ${L}^{2}\left(R\right)$ onto the subspace $\mathbf{H}$ of square integrable functions harmonic in $\mathbf{R}$. This problem is solved by decomposition $\mathbf{H}$ into the closed direct (not orthogonal) sum of two subspaces ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections ${P}^{\left(1\right)}$, ${P}^{\left(2\right)}$ of ${L}^{2}\left(R\right)$ onto ${\mathbf{H}}^{\left(1\right)}$, ${\mathbf{H}}^{\left(2\right)}$ respectively. The resulting construction...

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation...

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical...