A Dichotomy of PDE Sturmian Theory

Abstract

Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case do not extend to the hyperbolic case. Extensions to singular boundaries and mixed problems are included. The dichotomy referred to in the title pertains to the different but parallel Sturmian theories for elliptic and hyperbolic boundary problems: Results of similar type are true for both types of problems and yet other results true in the elliptic case are false in the hyperbolic case.