Radial conditions for global compact finite propagation for divergence-form elliptic operators
Abstract
We study global compact finite propagation (GCFP) for the wave equation utt=div(A(x)∇ u) on ℝd, that is, the property that a disturbance starting in a compact set cannot reach spatial infinity in finite time. For smooth strictly positive matrices A we show that GCFP implies ∫1∞ a*(r)−1/2 dr=∞, where a*(r) is the minimum over the sphere of radius r of ⟨ Aω,ω⟩; the proof uses the equivalence of GCFP with completeness of the Riemannian metric A−1, of which we prove the necessary direction for the cosine family of the Friedrichs extension. We also prove that the analogous divergence condition with a*(r), the maximum over the sphere, is sufficient, and that this holds for measurable coefficients A∈ L1loc with A≥ I. Finally, we construct a locally bounded measurable matrix with a*=a* for which GCFP holds although the integral converges, so the necessary condition is not stable under the loss of continuity.
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BibTeX
@misc{perelmuter2026radial,
title = {{Radial conditions for global compact finite propagation for divergence-form elliptic operators}},
author = {M. A. Perelmuter},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00876},
eprint = {2611.00876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00876},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial conditions for global compact finite propagation for divergence-form elliptic operators
AU - M. A. Perelmuter
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00876
UR - https://arxiv.org/abs/2611.00876
AB - We study global compact finite propagation (GCFP) for the wave equation u_tt=div(A(x)∇ u) on ℝ^d, that is, the property that a disturbance starting in a compact set cannot reach spatial infinity in finite time. For smooth strictly positive matrices A we show that GCFP implies ∫_1^∞ a_*(r)^−1/2 dr=∞, where a_*(r) is the minimum over the sphere of radius r of ⟨ Aω,ω⟩; the proof uses the equivalence of GCFP with completeness of the Riemannian metric A^−1, of which we prove the necessary direction for the cosine family of the Friedrichs extension. We also prove that the analogous divergence condition with a^*(r), the maximum over the sphere, is sufficient, and that this holds for measurable coefficients A∈ L^1_loc with A≥ I. Finally, we construct a locally bounded measurable matrix with a_*=a^* for which GCFP holds although the integral converges, so the necessary condition is not stable under the loss of continuity.
ER -