TY - GEN
T1 - High Frequency Weighted Resolvent Estimates for the Dirichlet Laplacian in the Exterior Domain
AU - Georgiev, Vladimir
AU - Rastrelli, Mario
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
PY - 2024
Y1 - 2024
N2 - In this paper, we want to present several resolvent estimates for the Dirichlet Laplacian in exterior domain. The estimates evaluate a weighted L2 norm with a weight measured by a negative power of the distance from the boundary. We consider an exterior domain Ω, that is the complementary of a compact in Rn, and the inhomogeneous Helmotz equation on it. If the exterior domain is non-trapping, there are cut-off resolvent estimates without weights. Our main result is that we can improve the estimates putting the weights. The main idea is the polar change of coordinates, where r=d(x,∂Ω), that allows us to use the Hardy inequality close to the boundary of the domain. Kato smoothing estimate is obtained as a consequence of the weighted cut-off resolvent estimates.
AB - In this paper, we want to present several resolvent estimates for the Dirichlet Laplacian in exterior domain. The estimates evaluate a weighted L2 norm with a weight measured by a negative power of the distance from the boundary. We consider an exterior domain Ω, that is the complementary of a compact in Rn, and the inhomogeneous Helmotz equation on it. If the exterior domain is non-trapping, there are cut-off resolvent estimates without weights. Our main result is that we can improve the estimates putting the weights. The main idea is the polar change of coordinates, where r=d(x,∂Ω), that allows us to use the Hardy inequality close to the boundary of the domain. Kato smoothing estimate is obtained as a consequence of the weighted cut-off resolvent estimates.
KW - Hardy inequalities
KW - Laplace-Betrami operator
KW - Resolvent estimates
UR - http://www.scopus.com/inward/record.url?scp=85198411221&partnerID=8YFLogxK
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U2 - 10.1007/978-3-031-53212-2_9
DO - 10.1007/978-3-031-53212-2_9
M3 - Conference contribution
AN - SCOPUS:85198411221
SN - 9783031532115
T3 - Springer Proceedings in Mathematics and Statistics
SP - 107
EP - 117
BT - New Trends in the Applications of Differential Equations in Sciences - NTADES 2023
A2 - Slavova, Angela
PB - Springer
T2 - 10th International Conference on New Trends in the Applications of Differential Equations in Sciences, NTADES 2023
Y2 - 17 July 2023 through 20 July 2023
ER -