Identification of a core from boundary data. (English) Zbl 0829.31004

Ring, Wolfgang

SIAM J. Appl. Math. 55, No. 3, 677-706 (1995).

The paper studies the problem of determining a continuous function \((r = a (\varphi) < 1\) (the interface) in the boundary value problem \[ \Delta u = \begin{cases} \mu_1 \quad & \text{for } 0 \leq r < a (\varphi) \\ \mu_2 \quad & \text{for } a (\varphi) \leq r \end{cases} \] on the unit disk \(U = \{r < 1\}\) from known boundary data \(z_1 = u |_{\partial U}\) and \(z_2 = {\partial u \over \partial u} |_{\partial U}\); here \(\mu_1\) and \(\mu_2\) are distinct positive numbers. The problem is shown to be equivalent to a nonlinear integral equation \(F(a) = z(z_1, z_2)\). It is analyzed and implemented by means of Fourier analysis. The equation is reduced to an infinite system of nonlinear functionals on the function \(a (\varphi)\), attaining prescribed values. Some results on smoothness, injectivity and ill-posedness are obtained. The author also presents numerical results for different interfaces.

Reviewer: A.Yu.Rashkovsky (Khar’kov)

MSC:

31B20	Boundary value and inverse problems for harmonic functions in higher dimensions
35R30	Inverse problems for PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R25	Ill-posed problems for PDEs

Keywords:

inverse source problem; Fourier transform; nonlinear integral equation; infinite system of nonlinear functionals; smoothness; injectivity; ill- posedness; numerical results