Inverse problems for linear and semilinear elliptic equations of the Schroedinger type

In this dissertation we study the inverse problems for elliptic partial differential equations of second order. We focus on the Schrodinger operator with complex coefficients. The problems discussed in this paper include inverse boundary value problems and inverse scattering problems. We study the problems in the cases of linear and semilinear equations in 2-dimension and higher dimensions.; In Chapter 2, we use integral equation method to construct the exponentially growing solutions of a class of elliptic equation of the Schrodinger type.; In Chapter 3, we establish an identity and the asymptotic of the exponentially growing solutions we obtained in Chapter 2, and use them to solve the inverse boundary value problem of linear equations in 3 and higher dimensions. We obtain the uniqueness (up to a transformation) in determining the coefficients of the equation from the Dirichlet to Neumann map (at the boundary).; In Chapter 4, we deal with semilinear equations by linearization. We also obtain a uniqueness result for a class of semilinear equations.; In Chapter 5, we study the 2-dimensional case which is known to be formally determined. And we study the inverse scattering problem which is closely related to the inverse boundary value problems. The results of Chapter 5 help us obtain information of the potential functions from the scattering amplitude.