On The Well Posedness Of The Inverse Coefficient Problems Of Two Kinds Of Equations

In this paper,the coefficient inverse problem of elliptic and parabolic equations are studied,and the two kinds of inverse problems in physics,financial,medical,geological detection and wireless transmission of electromagnetic field or in metal forming technology has a very wide range of applications.The first part,convexity of the corresponding functionals of the elliptic equation is proved based on Frechet derivative,getting the uniqueness of the solution can be determined by the functional properties;in the second part,the mathematical model is a parabolic equation,and it explores the optimization problem of the original problem of the corresponding solutions well posedness by using the optimal control theory.The difficulty of this problem is that: first,the inversion required two order parabolic equation is one of the two order coefficients,which is known as a fully nonlinear and severely ill-posed problem;at the same time,the additional conditions given is not usual sense of value,rather than the value of the integral mean observation,and the additional conditions of this type will cause the necessary conditions control functional minimizer must meet the extremely complex.Furthermore,it is difficult to obtain the uniqueness of the optimal solution because the control function is not convex.When the terminal point T is relatively small,we can prove the local uniqueness and stability of the minimizer by carefully analyzing the necessary condition and using the priori estimates of the forword problem,which is also the main work.This paper mainly includes the following four parts:The first part introduces the background of the inverse problem,the research situation at home and abroad,and the process of the development of the inverse problem.The first chapter theoretically analyzes a coefficient inverse problem of elliptic equation;we first introduce the mathematical model of the elliptic equation,and the inverse problem is ill-posed,so we need to estimate the positive problem of elliptic equation by using the theory of the frechet partial derivative equation estimation formula of deformation energy,to obtain the convexity of functional corresponding to elliptic equation.Due to the nature of the functional can get the uniqueness of solutions for semilinear elliptic equations.The second chapter studies diffusion coefficient of the parabolic equation inverse problem,and it is transformed into an optimal control problem P because of the ill-posed of the original problem,to use the optimization method to study the inverse problem of diffusion coefficient.According to the energy estimation of the positive problem and the energy estimation of the corresponding conjugate equation,then,the necessary conditions of the optimal solution are obtained by using the energy estimate.Finally,the uniqueness and stability of the optimal solution are obtained when the terminal point T is relatively small.In the third chapter,we summarize the future work of the inverse problems of parabolic and elliptic equations.