The ILL-POSEDNESS OF THE MIXED PROBLEM IN THE CYLINDRICAL DOMAIN FOR THE MULTIDIMENSIONAL LAVRENTIEV-BITSADZE EQUATION

Abstract

Studies of well-posed and ill-posed problems in mathematical physics, including inverse problems  and their practical applications, are of considerable interest, where the key issue is the correct formulation of the direct problem. Hyperbolic and elliptic equations are widely used in biomedical modeling, including to describe tumor growth and deformations of biological tissues. Analogies between membrane oscillations and tissue dynamics are widely used in biomechanics and mathematical medicine. For example, the spatial oscillations of elastic membranes are described
by partial differential equations. When the membrane deflection is specified by a function u(x,t), x ∈ R^m, m ≥ 2, application of Hamilton’s principle leads to a multidimensional wave equation, and in the case of equilibrium, to the Laplace equation. Consequently, the dynamics of elastic membranes can be described by the multidimensional Lavrentiev-Bitsadze equation. The problems considered in the article are ill-posed problems. The proof of non-unique solvability and the construction of an explicit solution is in fact a regularization of an ill-posed problem through the spectral method and integral representations, etc. In this article, the ambiguity of the solution is proven and an explicit form of the classical solution of a mixed problem for the multidimensional Lavrentiev-Bitsadze equation, is presented.