A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Abstract

Holmgren's theorem guarantees unique continuation across non-characteristic surfaces of class C ¹ for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C ¹-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T ₀. The minimal time T ₀ depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwell's equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C ¹-coefficients as long as they satisfy the conclusion of Holmgren's Theorem.

Keywords:

• unique continuation
• hyperbolic systems
• Holmgren's theorem

AMS Subject Classification:

• 35

Acknowledgement

Research of D. Toundykov is supported by the National Science Foundation through grant DMS-0908270