A quantified interface identity for boundary stabilization of Zaremba problems in dimension two
Abstract
We study the boundary stabilization of the planar variable-coefficient wave equation utt=div(A∇ u) under mixed Zaremba boundary conditions, the interface Γ=ΓD̅∩ΓN̅ being a finite set of points. For strong solutions, the flux of the Riemannian multiplier through a small half-circle centred at each point of Γ converges to −π/8⟨ H,τ⟩g η2, where η is the coefficient of the Shamir singularity and τ the unit tangent to the boundary pointing from the Neumann face to the Dirichlet face; this term has a favourable sign under ⟨ H,τ⟩g≤ 0. The identity extends, in an intrinsic and normalized form for anisotropic variable coefficients, the interface identities of Bey–Lohéac–Moussaoui and of Cornilleau–Lohéac–Osses. Under pointwise geometric assumptions, a Lyapunov functional yields exponential decay of the energy with named constants and without unique continuation. The edge decomposition is fully proved, by Mellin transform and dyadic weighted estimates. Under strict transversality, the time-integrated edge coefficient is controlled by the initial energy and extends continuously to all finite-energy data, as an element of L2(0,∞;L2(Γ)). A smooth example (the disc with a Dirichlet arc and a conformal metric of negative curvature) satisfies all assumptions with explicit constants. The case n≥ 3 is treated in a companion paper.
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BibTeX
@misc{bey2026quantified,
title = {{A quantified interface identity for boundary stabilization of Zaremba problems in dimension two}},
author = {Jean-Paul Bey},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05689295v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A quantified interface identity for boundary stabilization of Zaremba problems in dimension two
AU - Jean-Paul Bey
PY - 2026
DA - 2026/10/20
PB - HAL
UR - https://hal.science/hal-05689295v2
AB - We study the boundary stabilization of the planar variable-coefficient wave equation u_tt=div(A∇ u) under mixed Zaremba boundary conditions, the interface Γ=Γ_D̅∩Γ_N̅ being a finite set of points. For strong solutions, the flux of the Riemannian multiplier through a small half-circle centred at each point of Γ converges to −π/8⟨ H,τ⟩_g η^2, where η is the coefficient of the Shamir singularity and τ the unit tangent to the boundary pointing from the Neumann face to the Dirichlet face; this term has a favourable sign under ⟨ H,τ⟩_g≤ 0. The identity extends, in an intrinsic and normalized form for anisotropic variable coefficients, the interface identities of Bey–Lohéac–Moussaoui and of Cornilleau–Lohéac–Osses. Under pointwise geometric assumptions, a Lyapunov functional yields exponential decay of the energy with named constants and without unique continuation. The edge decomposition is fully proved, by Mellin transform and dyadic weighted estimates. Under strict transversality, the time-integrated edge coefficient is controlled by the initial energy and extends continuously to all finite-energy data, as an element of L^2(0,∞;L^2(Γ)). A smooth example (the disc with a Dirichlet arc and a conformal metric of negative curvature) satisfies all assumptions with explicit constants. The case n≥ 3 is treated in a companion paper.
ER -