In this dissertation we obtain new Carleman formulas for the solution of the Cauchy problem for equations P u = h, in Ω, u|E = f , where E ⊂ ∂Ω and | E | > 0. Our elliptic operator is of the form P = [ [2∂¯ 0 ; 0 2δ ] + A (x), where A is a 2 x 2 matrix. We also obtain estimates for the solution of equation P u = 0 when u is given at a finite number of points, and we prove that non-trivial solutions to the equation can not be small on large portions of the boundary, | Eδ | ≤ c / ln δ−1 , δ ∈ (0, 1), where Eδ = {z ∈ ∂Ω| | u(z) | < δ} and | Eδ | is the Lebesgue measure of Eδ . Finding the boundary condition from only a finite number of interior measurements of a domain is interesting both theoretically and practically. For example, when the boundary is physically inaccessible, all measurements must be made within the domain itself, and the conditions on the boundary must be reconstructed. We investigate the problem of recovering a boundary condition of the third kind for the Laplace operator defined on a simply connected domain in the complex plane, when the value of the solution and its gradient are known only for a finite number of interior points.

Thesis (Ph.D.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics