Сorrectness of the Dirichlet problem in the multidimensional domain for the model equations of mixed type

Author(s):  S.A. Aldashev, Dr., Prof., Kazakh National Pedagogical University named after Abai, Almaty, Kazakhstan , аldash51@mail.ru

Issue:  Volume 37, №1

Rubric:  Mathematics

Annotation:  It is known that the vibrations of elastic membranes in space are modeled by partial differential equations. If the deflection of the membrane is considered a function 1 ( , ), = ( ,..., ), 2, m uxt x x x m ≥ then by the Hamilton principle we arrive at multidimensional hyperbolic equations. Assuming that the membrane is in equilibrium in the bending position, the Hamiltonian principle also yields multidimensional elliptic equations. Consequently, the vibrations of elastic membranes in space can be modeled as multidimensional hyperbola-elliptic equations. The problem of the well-posedness of the Dirichlet problem for mixed-type equations in special domains was the subject of research by many authors on the plane and space. The method used in the author's papers is used to demonstrate unambiguous solvability and obtain an explicit form of the classical solution of the Dirichlet problem in the multidimensional region for the Lavrent'ev-Bitsadze equation

Keywords:  multidimensional domain, Dirichlet problem, unique solvability, spherical functions, orthogonality

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