【作者】 徐定华；

【导师】 李明忠； 黄思训；

【作者基本信息】 上海大学 ， 计算数学， 2001， 博士

【摘要】 本文主要围绕数学物理中的若干类反问题与边值问题进行研究，包括Hausdorff矩问题、弹性接触反问题、热传导方程反问题、Laplace方程Cauchy问题和二阶椭圆型方程组非线形边值问题等。随着科学技术的纵深发展和社会与经济的全面进步，越来越多的问题可归结为上述问题，如地质勘察、无损探伤、CT技术、军事侦察、环境治理、遥感遥测、信号处理、控制论、经济学等。本文创造性地应用积分方程方法，借助现代数学手段，着重研究这些问题的可解性条件、条件适定性(特别是条件稳定性)，构造稳定化算法，并进行数值模拟。论文的主要成果包括： 利用积分方程方法讨论通过有限个矩量求解Hausdorff矩问题，我们先将矩问题化为第一类Fredholm积分方程，分析该问题的不适定性。由于问题的不适定性，求解变得非常困难。为克服困难，我们先讨论矩问题的条件稳定性(包括整体估计和局部估计)，创造性地获得了对数型稳定性结果。并且基于稳定性，结合Tikhonov正则化方法，构造稳定化算法，成功地获得了矩问题的近似解——正则化解与精确解的整体误差估计和局部误差估计。利用有限元方法(有限维空间逼近)进行离散和数值模拟，数值实验结果很好地验证了算法的稳定性和有效性。本文提出的矩问题的稳定化算法可用之于Laplace方程Cauchy问题和一些反问题的数值求解。 在弹性理论中，接触反问题是一类重要的问题，属于一2001年上梅大学博士学位论文类无损探伤间题.我们首先给出了接触反问题的一种合理提法，即通过接触域外部的位移测量值决定不可直接测量的接触域和接触域上的应力.从数学上证明了接触反问题提法的合理性，第一次提出了接触反问题，并证明了其不适定性.为决定接触域和接触域上的应力，必须克服不适定性，构造稳定化算法，我们的基本思路是:先通过Fourier变换将接触反问题等价地转化为第一类Redhofm积分方程，再讨论积分方程解的唯一性条件和条件稳定性，从而通过位势理论创造性地获得了接触反问题的唯一性和对数型条件稳定性，包括整体和局部条件稳定性.最后构造反问题的Ti比onov正则化解(近似解)，利用稳定化估计证明了近似解的对数型整体误差估计和局部误差估计，我们的方法独具创新，为其他接触反问题的求解指明了方向. 本文讨论了高维情形下热传导方程中初始源项和中间某时刻温度分布的决定问题，证明了热传导方程反间题的合理提法，即通过部分边界或部分内部区域上、在有限时间内的温度分布值决定温度的初始分布和中间某一时刻温度分布.分别利用Carlemann估计、精确控制理论获得了初始温度分布决定的对数性稳定性估计、中间时刻温度分布的Lipschitz型稳定性.基于稳定性估计和正则化参数的先验选取，构造了正则化算法，获得了稳定化解，证明了该稳定化解的收敛速率. 利用小波分析工具讨论了助place方程Cauchy问题的稳定化求解，在小波多尺度分析中构造稳定化算法，获得了近似解的H说der型误差估计. 利用积分算子理论、奇异积分方程理论，讨论了二阶椭圆型方程组非线形边值问题(E2类和非瓦类)，建立了与间题等价的 数学物理中反问题与边值问题的积分方程方法奇异积分方程，并进而得到了问题的可解性条件及存在性证明. 为便于阅读，在本文开始还提纲掣领地叙述了反问题与边值问题的基本思想与方法，本文结束时，对本文进行了总结和评论，并对将来数学物理反问题和边值问题的研究进行了展望.更多还原

【Abstract】 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更多还原