The criterion of unique solvability of the dirichlet spectral problem in the cylindrical domain for a class of multi-dimensional hyperbolic-elliptic equations

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to the Hamilton principle can be modeled by multidimensional hyperbolic equations. Assuming that the membrane is in equilibrium in the bending position, Hamilton’s principle also yields multidimensional elliptic equations. Consequently, oscillations of elastic membranes in space can be modeled by multidimensional hyperbolic-elliptic equations. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, essentially depends on the height of the entire cylindrical region under consideration. Two-dimensional spectral problems for equations of the hyperbolic-elliptic type are intensively studied, however, as far as is known, their multidimensional analogs are poorly studied. In this paper, we obtain a criterion for the unique solvability of the Dirichlet spectral problem in a cylindrical domain for a class of multidimensional hyperbolicelliptic equations.