Abstract:
Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that vibrations of elastic membranes in space according to Hamilton's principle can be modeled by a multidimensional wave equation. Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation fromHamilton's principle. Thus, vibrations of elastic membranes in space can bemodeled as a multidimensional Lavrent'ev-Bitsadze equation. The theory of boundary value problems for hyperbolic-elliptic equations on a plane is wellexplored, and their multidimensional analogues are intensively analyzed. Two-dimensional spectral problems for equations of hyperbolic-elliptic type have been well researched, but their multidimensional analogues have been relatively under-studied. In this paper, we consider the spectral mixed problem for the multidimensional Lavrent'ev-Bitsadze equation and establish a criterion for its unique solvability. We also determine the eigenvalues and the corresponding eigenfunctions of this problem.
