Inverse Coefficient Problems For Nonlinear Mathematical Physics Equations

This dissertation investigates several types of inverse coefficient problems for nonlinear mathematical physics equations. Using the variational methods, we obtain the existence and uniqueness of weak solutions for nonlinear mathematical physics equations and verify the convergence of iterative approximation solutions. Meanwhile, we discuss the inverse coefficient problems via several analytical methods of inverse problems.the quasi-solution of the inverse coefficient problem is constructed in the appropriate class of admissible coefficients. Under some mild conditions, we analysis the existence, uniqueness, and stability of the solutions for the inverse coefficient problems.The first chapter presents basic notions and research significance for inverse problems in equations of mathematical physics. Some useful results in Sobolev space and nonlinear functional analysis are also summarized here.In chapter 2, we discuss a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of this class of problems depends on the gradient of its solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities is unique solvable for the given class of coefficients. The existence results of quasi solutions for the inverse problems are obtained.In chapter 3, we focus on the inverse coefficient problems for two equations of parabolic type. Under some suitable conditions, we obtain the existence, uniqueness, and stability of solutions for the inverse coefficient problems, in the case that an additional observation condition for the solutions is given.In chapter 4, we discuss the inverse coefficient problem for nonlinear potential operator equation. The unknown leading coefficient not only depends on the gradient of the solution in the equation but also is determined only by two measurement data. We prove that an existence and uniqueness theorem of weak solutions of direct problem by using the monotone potential operator theories and Browder-Minty theorem. The quasi-solution of the inverse coefficient problem is obtained in the appropriate class of admissible coefficients.In chapter 5, we consider nonlinear inverse problems in the framework of abstract operator equations in the form: F(x)=y. A new Newton-Landweber iteration is presented to solve this nonlinear equation. Under some proper conditions, we get the convergence.In chapter 6, using the variational methods and the theory of monotone operator, we study a few kinds of nonlinear boundary value problems, and we obtain the existence and uniqueness of weak solutions.