In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is Laplacian and nonlinear divergence of a function analytic gradient solution. The main novelty terms result that coefficients nonlinearity are allowed to be “anisotropic”. As previous works, proof reduces an integral identity involving tensor product gradients 3 or more...

An ill-posed Cauchy problem for a 3D elliptic PDE with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability error estimates are given. For the numerical implementation, a large sparse system is obtained from the discretizing the QBV problem using finite difference method (FDM). A LeftPreconditioner Generalized Minimum Re...

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express th...

We show that in dimension two or greater, a certain equivalence class of the scalar coefficient a(x, u) of the semilinear elliptic equation ∆u + a(x, u) = 0 is uniquely determined by the Dirichlet to Neumann map of the equation on a bounded domain with smooth boundary. We also show that the coefficient a(x, u) can be determined by the Dirichlet to Neumann map under some additional hypotheses.

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an innnite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonli...