半平面中的Hilbert边值逆问题

Inverse Hilbert Boundary Value Problems in Half Plane

边值问题逆问题是在边值问题中涉及到参变未知函数,它具有重要的力学背景,但对边值问题逆问题的研究才起步.从数学上给出半平面中解析函数的一类Hilbert边值逆问题的合理提法,将其转化为实轴上的解析函数的Riemann边值问题,依据实轴上解析函数Riemann边值问题的经典理论,讨论了半平面中解析函数的一类Hilbert边值逆问题的可解性,得到了该边值逆问题的解由该边值逆问题标数所决定的实的自由度,给出了该边值问题逆问题的可解条件和解的积分表达式.

Inverse problems are boundary value problems with parametric unknown functions.They possess important mechanical backgrounds.The study on the inverse boundary value problems is on the primary stage.In this paper,the mathematical formulation of a class of inverse Hilbert boundary value problems in half plane for analytic functions is given.On the basis of the classical theory of Riemann boundary value problems for analytic functions on real axis,the solvability of the inverse Hilbert boundary value problems is discussed by transforming them into Riemann boundary value problems.The real freedom of the solutions determined by the index of the inverse Hilbert boundary value problems is obtained.The solvable conditions and the integral representation of the solutions of the inverse Hilbert boundary value problems are also obtained.