Weyl formula for the eigenvalues of the dissipative acoustic operator

2021/04/06 by Vesselin Petkov, Petkov, Vesselin
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2104.02341

Abstract

We study the wave equation in the exterior of a bounded domain K with dissipative boundary condition ∂_ν u - γ(x) ∂_t u = 0 on the boundary Γ and γ(x) > 0. The solutions are described by a contraction semigroup V(t) = e^tG, t ≥ 0. The eigenvalues λ_k of G with \rm Re λ_k < 0 yield asymptotically disappearing solutions u(t, x) = e^{λ_k t} f(x) having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case min_{x∈ Γ} γ(x) > 1. For strictly convex obstacles K this formula concerns all eigenvalues of G.

Related

• Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles
• Location and Weyl formula for the eigenvalues of some non self-adjoint operators
• Location of eigenvalues for the wave equation with dissipative boundary conditions
• Asymptotic of the dissipative eigenvalues of Maxwell's equations
• Weyl formula for the negative dissipative eigenvalues of Maxwell's\n equations