Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation

Abstract

It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.