The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

IF 0.9 4区 数学 Q2 MATHEMATICS
Alexander V. Rogovoy, Tynysbek S. Kalmenov, Sergey I. Kabanikhin
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引用次数: 0

Abstract

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.
双曲盖勒斯特方程的超定考希问题
超定边界值问题及其产生的极小算子在微分方程规则边界值问题的描述中极为重要,在解的局部性质研究中也得到广泛应用。此外,对于应用中产生的数学物理反问题,在确定未知数据时,有必要研究具有超定边界条件的问题,这体现在对物理学、地球物理学、地震层析成像、地球电学、电动力学、医学、生态学、经济学和许多其他实际领域中产生的问题(包括双曲方程和系统)的研究中。因此,研究超定边界值问题既有理论意义,又有应用价值。本文建立了双曲域中 Gellerstedt 方程的过定 Cauchy 问题及其产生的最小微分算子的正则可解性准则,并考虑了特征三角形情况和对域边界有相当一般假设的更一般域情况。由于存在过度确定的边界条件,所考虑的问题在一般情况下是求解困难的,因此,为了使问题能够正常求解,必须对初始数据施加额外的条件。换句话说,我们考虑的是反问题:确定问题的初始数据,特别是盖勒斯特方程的右边部分,应该满足哪些要求,从而使过确定考西问题可以有规律地求解。证明的基础是盖勒斯特势、特征三角形中 Goursat 问题解的性质以及特殊函数的性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Inverse and Ill-Posed Problems
Journal of Inverse and Ill-Posed Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.60
自引率
9.10%
发文量
48
审稿时长
>12 weeks
期刊介绍: This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published. Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest. The following topics are covered: Inverse problems existence and uniqueness theorems stability estimates optimization and identification problems numerical methods Ill-posed problems regularization theory operator equations integral geometry Applications inverse problems in geophysics, electrodynamics and acoustics inverse problems in ecology inverse and ill-posed problems in medicine mathematical problems of tomography
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