Spherical Means of Solutions of Partial Differential Equations in a Conical Region

The spherical means of the solutions of a linear partial differential equation $Lu = f$ in a conical region are studied. The conical region is bounded by a surface generated by curvilinear $\xi $ lines and by two truncating $\xi $ surfaces. The spherical mean is the average of u over a constant $\xi $ surface. Conditions on the linear differential operator, L, and on the orthogonal coordinates $\xi $, $\eta $ ,$\xi $ are established so that the problem for the determination of the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be reduced to a problem with only one space variable. Conditions are then established so that the spherical mean of the solution in one conical region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.