Integral Equation Methods For Inverse Problems And Boundary Value Problems In Mathematical Physics

The dissertation discusses several classes of inverse problems and boundary value problems in mathematical physics, including Hausdorff moment problems, inverse contact problem in the theory of elasticity, inverse heat conductivity problem, Cauchy problem for Laplace equations, and non-linear boundary value problems for second-order elliptic differential systems. With the in-depth development in science and technology and all-around progress in society and economy, more and more practical problems, such as geological prospecting, non-destructive testing, CT technology, military reconnaissance, environmental disposal, remote sensing, signal processing, cybernetics, economics and so on, have been formulated into inverse problems and boundary value problems for partial differential equations. By means of integral equation methods creatively along with other modern mathematical theories, this paper focuses on finding solvability conditions and conditional well-posedness (especially conditional stability), constructing stabilized algorithms, and carrying through numerical simulation. The main achievements are as follows:Using integral equation methods to solve the Hausdorff moment problems (HMP) by finite moments, we first transform the HMP into the first kind of Fredholm integral equation equivalently and analyze the ill-posedness of the HMP, for which it is very difficult to solve. In order to cure the ill-posedness, we first discuss the conditional stability estimates in moment problems including global and local estimates, and have obtained the stability results of logarithmic rate creatively. Basing on the stability estimates and the famous Tikhonov regularization method, we present some stabilized algorithms and have successfully derived the error estimatesbetween the approximate solution (regularized solution) and the exact solution for the HMP, including the global and local error estimates. We discretize the algorithms by means of the finite element method and do some numerical simulations. The numerical results show the nice stability and efficiency of the algorithms. The presented algorithms here for the HMP are applicable to numerically solving Cauchy problems for Laplace equations and several inverse problems.Inverse contact problems, as a class of non-destructive problems, are important issues in the theory of elasticity. Firstly we give an appropriate presentation for an inverse contact problem, that is, we determine the inaccessible contact domain and the stress on the domain from the displacement measurements outside of the contact domain. We have formulated the inverse contact problem for the first time, and proved mathematically its rationality and its ill-posedness. In order to determine the contact domain and the stress on the domain, we must cure the ill-posedness and construct stabilized algorithms. Our main ideas are as follows: firstly we transform equivalently the inverse contact problem into the Fredholm integral equation of the first kind by the Fourier transforms, and discuss the uniqueness and conditional stability for the integral equation, and then by the potential theory we derive the uniqueness and stability estimates creatively, including the global and local stability estimates, finally we construct the Tikhonov regularized solution for the inverse contact problem and prove the global and local error estimates of logarithmic rate. Our method shows the originality and gives some instructive ideas for solving other inverse contact problems.The dissertation also discusses multi-dimensional inverse heat conductivity problems(IHCP), that is, the determination of initial source terms and heat distribution at any intermediate moment. We have proved the reasonable presentation of the IHCP, i.e., we can uniquely determine theheat distribution at the initial moment and at any moments from the heat distribution measurements on the part of the boundary or in arbitrary accessible sub-domain at any time-interval. Utilizing the Carlemann estimates and exact controllability theory res...