Interface identity and boundary stabilization of Zaremba problems in dimension n ≥ 3
Abstract
We study the boundary stabilization of the variable-coefficient wave equation utt=div(A∇ u) under mixed Zaremba boundary conditions in dimension n≥ 3, the interface Γ being a compact submanifold of codimension two across which the boundary condition switches from Dirichlet to Neumann along a smooth boundary. We first establish an intrinsic edge decomposition: near Γ, every strong solution reads u=uR+𝒦Γη+w, where 𝒦Γη=χ(r) r1/2sin(θ/2) ω(rΛ)η is defined through the functional calculus of the Laplacian of Γ and the remainder w has vanishing renormalized traces on tubes around Γ; this form is derived from the edge asymptotics of Dines–Harutjunjan–Schulze by strong convergence of traces, without identifying operators. We deduce the interface flux identity limε→ 0 Iε=−π/8∫Γ⟨ H,τ⟩g η2 for the Riemannian multiplier H, and exponential energy decay with named constants under pointwise geometric assumptions, the Lyapunov closure proved in dimension two in a companion paper applying without change. We add a Riemannian star-shapedness criterion at the interface, a sufficient condition for the existence of an escape field ∇g f+K (K Killing) compatible with the partition and, under strict transversality, the hidden regularity η∈ L2(0,∞;L2(Γ)) for all finite-energy data.
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BibTeX
@misc{bey2026interface,
title = {{Interface identity and boundary stabilization of Zaremba problems in dimension n ≥ 3}},
author = {Jean-Paul Bey},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05828301v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Interface identity and boundary stabilization of Zaremba problems in dimension n ≥ 3
AU - Jean-Paul Bey
PY - 2026
DA - 2026/10/20
PB - HAL
UR - https://hal.science/hal-05828301v1
AB - We study the boundary stabilization of the variable-coefficient wave equation u_tt=div(A∇ u) under mixed Zaremba boundary conditions in dimension n≥ 3, the interface Γ being a compact submanifold of codimension two across which the boundary condition switches from Dirichlet to Neumann along a smooth boundary. We first establish an intrinsic edge decomposition: near Γ, every strong solution reads u=u_R+𝒦_Γη+w, where 𝒦_Γη=χ(r) r^1/2sin(θ/2) ω(rΛ)η is defined through the functional calculus of the Laplacian of Γ and the remainder w has vanishing renormalized traces on tubes around Γ; this form is derived from the edge asymptotics of Dines–Harutjunjan–Schulze by strong convergence of traces, without identifying operators. We deduce the interface flux identity lim_ε→ 0 I_ε=−π/8∫_Γ⟨ H,τ⟩_g η^2 for the Riemannian multiplier H, and exponential energy decay with named constants under pointwise geometric assumptions, the Lyapunov closure proved in dimension two in a companion paper applying without change. We add a Riemannian star-shapedness criterion at the interface, a sufficient condition for the existence of an escape field ∇_g f+K (K Killing) compatible with the partition and, under strict transversality, the hidden regularity η∈ L^2(0,∞;L^2(Γ)) for all finite-energy data.
ER -