The correctness of a Dirichlet type problem in a cylindrical domain for the multidimensional Lavrentiev--Bitsadze equation

S. A. Aldashev

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 24 (2018) no. 1, pp. 7-13

Résumé

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical and geometric processes. It is known that vibrations of elastic membranes in space according to the Hamiltonian principle can be modeled by a multidimensional wave equation. Assuming that the membrane is in equilibrium in the bending position, the Hamiltonian principle also yields the multidimensional Laplace equation. Consequently, the vibrations of elastic membranes in space can be modeled as the multidimensional Lavrentiev–Bitsadze equation. When studying these applications, it becomes necessary to obtain an explicit representation of the boundary value problems being studied. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where a unique solvability of this problem is shown, which essentially depends on the height of the entire cylindrical region under consideration. In this paper we investigate a Dirichlet type problem in the cylindrical domain for the multidimensional Lavrentiev–Bitsadze equation and obtain an explicit form of its classical solution. In this case, the unique solvability depends only on the height of the hyperbolic part of the cylindrical domain, and a criterion for the uniqueness of the solution is given.

Détail

Keywords: well-posedness, Dirichlet type problem, cylindrical domain, multidimensional equation, criterion.