Proves that Weyl asymptotics of the Friedrichs Laplacian on singular metrics with variable boundary degeneracy α are governed by max α when above 2/(n+1) and by truncated volume otherwise, with explicit constants and logs when the max set is Morse-Bott.
Colin de Verdi` ere, C
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abstract
This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace-Beltrami operator is presented and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.
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Proves Pleijel-type asymptotic upper bound on nodal domains of eigenfunctions for a class of degenerate elliptic operators including Baouendi-Grushin, matching the Dirichlet Laplacian constant.
Observability inequality for waves on singular-boundary Riemannian manifolds is established by reducing the general case to a separable one via perturbation and applying uniform tangential-frequency analysis plus an Ingham inequality.
citing papers explorer
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Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent
Proves that Weyl asymptotics of the Friedrichs Laplacian on singular metrics with variable boundary degeneracy α are governed by max α when above 2/(n+1) and by truncated volume otherwise, with explicit constants and logs when the max set is Morse-Bott.
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Pleijel's theorem for a class of degenerate elliptic operators
Proves Pleijel-type asymptotic upper bound on nodal domains of eigenfunctions for a class of degenerate elliptic operators including Baouendi-Grushin, matching the Dirichlet Laplacian constant.
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Boundary observability for gas giant metrics
Observability inequality for waves on singular-boundary Riemannian manifolds is established by reducing the general case to a separable one via perturbation and applying uniform tangential-frequency analysis plus an Ingham inequality.