Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles

2023/10/02 by Vesselin Petkov, Petkov, Vesselin
Computer Science · Mathematics · #35P20 #35P25 #47A40 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2310.01192

openalex publication_date 2023/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We examine the wave equation in the exterior of a strictly convex bounded domain K with dissipative boundary condition ∂_ν u - γ(x) ∂_t u = 0 on the boundary Γ and 0 < γ(x) <1, ∀ x ∈ Γ. The solutions are described by a contraction semigroup V(t) = e^tG, t ≥ 0. The poles λ of the meromorphic incoming resolvent (G - λ)^-1: \mathcal H_comp → \mathcal D_loc are eigenvalues of G if \rm Re λ< 0 and incoming resonances if \rm Re λ> 0. We obtain sharper results for the location of the eigenvalues of G and incoming resonances in Λ= \λ∈ \mathbb C: |\rm Re λ| ≤ C₂(1 + |\rm Im λ|)^-2, |\rm Im λ| ≥ A₂ > 1\ and we prove a Weyl formula for their asymptotic. For K = \x ∈ \mathbb R³: |x| ≤ 1\ and γ constant we show that G has no eigenvalues so the Weyl formula concerns only the incoming resonances.

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