A DOUBLE LAYER POTENTIAL APPROACH FOR PLANAR CAUCHY PROBLEMS FOR THE LAPLACE EQUATION

Анотація

We derive a double-layer potential approach which recasts the ill-posed Cauchy problem for the Laplace equation in a planar bounded doubly-connected domain as a system of boundary integral equations. It is shown that the system has a unique solution for a dense set of data. For the integral equations involved having a hypersingular kernel the Nystr\"om method based on trigonometrical quadratures is applied. The resulting fully discrete linear system is solved by Tikhonov regularization. Results included of numerical experiments show that a stable approximation can be obtained with the double-layer method to the solution of the Cauchy problem.