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.

中文翻译：

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具有奇异系数的混合型方程无界域中并行特征数据的 Gellerstedt-Moiseev 问题

 抽象的

在这项工作中，在由半平面 \(y>0\) 和特征三角形 \(y<0\) 组成的无界域中，考虑具有奇异系数的椭圆双曲型简并方程方程的较低项。针对与其平行的部分边界和内部特征数据，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，使用狄利克雷问题解的积分表示。在特征三角形中，使用了达布公式，该公式给出了修正柯西问题的解的积分表示，其中数据位于 \(y=0\) 轴的线段 \([-1,1]\) 上。为了证明问题解的唯一性，采用了极值原理（对于专门构造的有限域\(D_{R}\)）和从有限域\(D_{R}到极限的传递)的组合方法\) 到无界域 \(D\) 被使用。利用狄利克雷和达布公式，GM问题解的存在性等价地简化为非标准奇异积分方程组的研究，其中非特征部分包含非Fredholm算子，其核为具有孤立的一阶奇点。利用卡尔曼方法，即暂时假设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。从获得的两个关系式中，其中一个未知函数可以通过第二个未知函数来明确表达，这使得可以将该系统简化为维纳-霍普夫积分方程，该方程属于奇异积分方程类。 已证明该方程的指数为零。通过求解该方程，得到第二类 Fredholm 积分方程。该方程的唯一可解性源自\(GM\) 问题解的唯一性。