Generalized Solutions of the Hamilton-Jacobi Equations of Eikonal Type. I. Formulation of the Problems; Existence, Uniqueness and Stability Theorems; Some Properties of the Solutions

A nonlocal theory of boundary value problems in bounded and unbounded domains is constructed for nonlinear equations of first order of the type of the classical eikonal equation in geometric optics with Cauchy-Dirichlet boundary conditions and with additional conditions at infinity in the case of unbounded domains. Existence, uniqueness and stability theorems are proved; some representations of the solutions are indicated; and properties of the solutions generalizing the principles of Fermat and Huygens in geometric optics are established. Bibliography: 14 titles.