Optimal constants in nontrapping resolvent estimates and applications in numerical analysis

Abstract

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds the outgoing resolvent satisfies $\parallel \chi R\left(k\right)\chi {\parallel }_{L^2\to L^2}\le C{k}^{-1}$ for $k>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $\frac{2}{\pi }$ times the length of the longest generalized bicharacteristic of $|\xi {|}_g^2-1$ remaining in the support of $\chi$. We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.