{"title":"Gellerstedt–Moiseev Problem with Data on Parallel Characteristics in the Unbounded Domain for a Mixed Type Equation with Singular Coefficients","authors":"A. K. Urinov, D. M. Mirsaburova","doi":"10.1134/s199508022460081x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this work, in an unbounded domain, which consists of a half-plane <span>\\(y>0\\)</span> and a characteristic triangle for <span>\\(y<0\\)</span>, a degenerate equation of elliptic-hyperbolic type with singular coefficients is considered for the lower terms of the equation. The correctness of the Gellerstedt–Moiseev (<span>\\(GM\\)</span>) problem is studied for data on the part of the boundary and internal characteristics parallel to it. When studying the <span>\\(GM\\)</span> problem in the half-plane <span>\\(y>0\\)</span>, the integral representation of the solution of the Dirichlet problem is used. In the characteristic triangle the Darboux formula, which gives an integral representation of the solution to the modified Cauchy problem with data on the segment <span>\\([-1,1]\\)</span> of the <span>\\(y=0\\)</span> axis, is used. To prove the uniqueness of the solution to the problem, a combined method of the extremum principle (for a specially constructed finite domain <span>\\(D_{R}\\)</span>) and the passing to the limit from the finite domain <span>\\(D_{R}\\)</span> to the unbounded domain <span>\\(D\\)</span> are used. Using the Dirichlet and Darboux formulas the existence of the solution to the <span>\\(GM\\)</span> problem is equivalently reduced to the study of the system of non-standard singular integral equations, which the non-characteristic parts contain non-Fredholm operators with kernels that have isolated first-order singularities. Using the Carleman’s method, i.e., temporarily assuming the non-characteristic parts of these equations as known functions, the regularization of these equations are carried out. From the obtained two relations, one of the unknown function is explicitly expressed through the second one and this makes it possible to reduce this system to the Wiener–Hopf integral equation, which belongs to the class of singular integral equations. It has been proved that the index of this equation is equal to zero. By solving this equation a second kind Fredholm integral equation is obtained. The uniquely solvability of this equation follows from the uniqueness of the solution of the <span>\\(GM\\)</span> problem.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s199508022460081x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this work, in an unbounded domain, which consists of a half-plane \(y>0\) and a characteristic triangle for \(y<0\), a degenerate equation of elliptic-hyperbolic type with singular coefficients is considered for the lower terms of the equation. The correctness of the Gellerstedt–Moiseev (\(GM\)) problem is studied for data on the part of the boundary and internal characteristics parallel to it. When studying the \(GM\) problem in the half-plane \(y>0\), the integral representation of the solution of the Dirichlet problem is used. In the characteristic triangle the Darboux formula, which gives an integral representation of the solution to the modified Cauchy problem with data on the segment \([-1,1]\) of the \(y=0\) axis, is used. To prove the uniqueness of the solution to the problem, a combined method of the extremum principle (for a specially constructed finite domain \(D_{R}\)) and the passing to the limit from the finite domain \(D_{R}\) to the unbounded domain \(D\) are used. Using the Dirichlet and Darboux formulas the existence of the solution to the \(GM\) problem is equivalently reduced to the study of the system of non-standard singular integral equations, which the non-characteristic parts contain non-Fredholm operators with kernels that have isolated first-order singularities. Using the Carleman’s method, i.e., temporarily assuming the non-characteristic parts of these equations as known functions, the regularization of these equations are carried out. From the obtained two relations, one of the unknown function is explicitly expressed through the second one and this makes it possible to reduce this system to the Wiener–Hopf integral equation, which belongs to the class of singular integral equations. It has been proved that the index of this equation is equal to zero. By solving this equation a second kind Fredholm integral equation is obtained. The uniquely solvability of this equation follows from the uniqueness of the solution of the \(GM\) problem.
AbstractIn this work, in an unbounded domain, which consists of a half-plane \(y>0\) and a characteristic triangle for \(y<0\), is considered a degenerate equation of elliptic-hyperbolic type with singular coefficients for the lower terms of the equation.对于边界部分的数据和与之平行的内部特征，研究了 Gellerstedt-Moiseev （\(GM\)）问题的正确性。在研究半平面 \(y>0\)中的\(GM\)问题时，使用了迪里夏特问题解的积分表示法。在特征三角形中，使用了达尔布公式，它给出了修正考希问题解的积分表示，其数据在（y=0）轴的（[-1,1]\）段上。为了证明问题解的唯一性，使用了极值原理（对于一个特殊构造的有限域 \(D_{R}\））和从有限域 \(D_{R}\)到无界域 \(D\)的极限传递的组合方法。利用狄利克雷公式和达尔布公式，\(GM\)问题解的存在性等价地简化为非标准奇异积分方程系统的研究，其中非特征部分包含非弗雷德霍姆算子，其核具有孤立的一阶奇异性。利用卡勒曼方法，即暂时假定这些方程的非特征部分为已知函数，对这些方程进行正则化。从得到的两个关系式中，一个未知函数通过第二个关系式明确表达出来，这使得将该系统简化为维纳-霍普夫积分方程成为可能，而后者属于奇异积分方程。事实证明，该方程的指数等于零。通过求解这个方程，可以得到一个第二类弗雷德霍姆积分方程。该方程的唯一可解性源于 \(GM\) 问题解的唯一性。
期刊介绍：
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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