A local energy estimate for 2-dimensional Dirichlet wave equations

Hepditch, Kellan; Metcalfe, Jason

doi:10.2140/involve.2023.16.483

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A local energy estimate for 2-dimensional Dirichlet wave equations

Kellan Hepditch and Jason Metcalfe

Vol. 16 (2023), No. 3, 483–492

DOI: 10.2140/involve.2023.16.483

Abstract

We examine a variant of the integrated local energy estimate for $(1 + 2)$ -dimensional Dirichlet wave equations exterior to star-shaped obstacles. The classical bound on the solution, rather than the derivative, is not typically available in two spatial dimensions. Using an argument inspired by the $r^{p}$ -weighted method of Dafermos and Rodnianski and taking advantage of the Dirichlet boundary conditions allow for the recovery of such a term when the initial energy is appropriately weighted.

Keywords

wave equation, local energy estimate, exterior domain

Mathematical Subject Classification

Primary: 35L71, 35L05

Milestones

Received: 7 February 2022

Revised: 15 June 2022

Accepted: 11 September 2022

Published: 10 August 2023

Communicated by Suzanne Lenhart

Authors

Kellan Hepditch
Cedar Ridge High School Hillsborough, NC United States

Jason Metcalfe
Department of Mathematics University of North Carolina Chapel Hill, NC United States

Vol. 16, No. 3, 2023