### Abstract

Original language | English |
---|---|

Journal | Pure and Applied Analysis |

Publication status | Accepted/In press - 26 Oct 2019 |

### Keywords

- math.AP
- math.NA

### Cite this

*Pure and Applied Analysis*.

**Optimal constants in nontrapping resolvent estimates and applications in numerical analysis.** / Galkowski, Jeffrey; Spence, Euan A.; Wunsch, Jared.

Research output: Contribution to journal › Article

*Pure and Applied Analysis*.

}

TY - JOUR

T1 - Optimal constants in nontrapping resolvent estimates and applications in numerical analysis

AU - Galkowski, Jeffrey

AU - Spence, Euan A.

AU - Wunsch, Jared

N1 - 40 pages

PY - 2019/10/26

Y1 - 2019/10/26

N2 - We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ 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.

AB - We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ 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.

KW - math.AP

KW - math.NA

M3 - Article

JO - Pure and Applied Analysis

JF - Pure and Applied Analysis

SN - 2578-5893

ER -