Об одной задаче типа Дирихле для уравнения составного типа

Abstract

Исследуется краевая задача для класса уравнений третьего порядка составного типа с эллиптическим оператором в главной части. Доказаны теоремы существования и единственности классического решения для рассматриваемых задач. Доказательство основано на энергетических неравенствах и на теории интегральных уравнений фредгольмовского типа. The aim of this paper is to prove the existence and uniqueness of smooth solutions to the Dirichlet type problem for one class of third-order equations that do not belong to any of the classic types. One of the main classes of non-classical equations is third-order composite type equations, the operator of which is a composition of first-order hyperbolic operator and an elliptic operator in the main part. A number of boundary value problems for the model composite type equations with the Laplace operator were investigated by T.D. Dzhuraev. Many studies have proved the existence of solutions to boundary value problems upon fulfillment of conditions of the convexity of area boundary. The method of proof used in this paper is similar to the method used in the research paper of the author mentioned above. For the research of composite type linear equations a combination of the method of potentials (Green's function) and integral identities is applied. The research method is based on reducing the studied problem with the help of the Green's function to an integral equation, the proof of its solubility and thus - the proof of the solvability of original problem. Upon fulfillment of certain conditions on given functions, a third-order equation reduces to a second-order equation of elliptic type with an unknown right-hand side and the boundary function. With the help of Green's functions for elliptic equations, the studied problem is reduced to a second-order equivalent integral equation, the solvability follows from Fredholm alternative and the theorem of uniqueness of the solution of the original problem.