A Direct Integral Equation Method for a Cauchy Problem for the Laplace Equation in 3-Dimensional Semi-Infinite Domains

摘要

We consider a Cauchy problem for the Laplace equation in a 3-dimensional semi-infinite domain that contains a bounded inclusion. The canonical situation is the upper half-space in IR~3 containing a bounded smooth domain. The function value of the solution is specified throughout the plane bounding the upper half-space, and the normal derivative is given only on a finite portion of this plane. The aim is to reconstruct the solution on the surface of the bounded inclusion. This is a generalisation of the situation in Chapko and Johansson (2008) to three-dimensions and with Cauchy data only partially given. We represent the solution in terms of a sum of a layer potential over the surface over the inclusion with an unknown density and a layer potential involving a Green's function and a known density (the given data on the plane). The Cauchy problem is then reduced to identifying the unknown density. To construct it, we match up the data on the finite portion of the plane, where both function values and the normal derivative are specified, and this gives rise to a integral equation of the first kind over the (bounded) surface of the inclusion having a smooth kernel. We show that this boundary integral equation is uniquely solvable for a certain class of data in the usual Sobolev and Holder type spaces. To numerically solve this equation, we employ Weinert's method [Wienert (1990)]. This involves rewriting the integral equation over the unit sphere under the assumption that the surface of the inclusion can be mapped one-to-one to the unit sphere. The density is then represented in terms of a linear combination of spherical harmonics, and this generates a linear system to solve for the coefficients in this representation. Due to the ill-posedness of the Cauchy problem, Tikhonov regularization is incorporated. Numerical results are given as well, showing that accurate reconstructions of the solution and its normal derivative can be obtained on the surface of the inclusion with small computational effort. We also investigate the case when the normal derivative is given throughout the plane and the function value is only specified at a finite portion, and compare the accuracy of the reconstructions.