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Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy
Abstract
We prove local rigidity of smooth conformally Euclidean metrics g0 on smooth compact domains M⊂ℝn, n≥3, for the anisotropic Helmholtz equation (−Δg −k2)u =0 at a fixed energy k2>0 which is not a Dirichlet eigenvalue of −Δg0 in M. Any smooth metric sufficiently close to g0 in Hs(M), with integer s>n/2+1, and having the same Dirichlet-to-Neumann map, is its pullback by a smooth diffeomorphism fixing the boundary pointwise. The proof uses a Helmholtz embedding to fix the gauge and a compactness argument together with a characterization of the kernel of the linearization.
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BibTeX
@misc{munozthon2026local,
title = {{Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy}},
author = {Sebastián Muñoz-Thon and Plamen Stefanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25314},
eprint = {2610.25314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25314},
}
RIS
TY - UNPB
TI - Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy
AU - Sebastián Muñoz-Thon
AU - Plamen Stefanov
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.25314
UR - https://arxiv.org/abs/2610.25314
AB - We prove local rigidity of smooth conformally Euclidean metrics g_0 on smooth compact domains M⊂ℝ^n, n≥3, for the anisotropic Helmholtz equation (−Δ_g −k^2)u =0 at a fixed energy k^2>0 which is not a Dirichlet eigenvalue of −Δ_g_0 in M. Any smooth metric sufficiently close to g_0 in H^s(M), with integer s>n/2+1, and having the same Dirichlet-to-Neumann map, is its pullback by a smooth diffeomorphism fixing the boundary pointwise. The proof uses a Helmholtz embedding to fix the gauge and a compactness argument together with a characterization of the kernel of the linearization.
ER -