Quantitative Inverse-Norm Theory for Robin Boundary Integral Operators
Quantitative Inverse-Norm Theory for Robin Boundary Integral Operators
Ming Yang1
作者信息
- 1. School of Mathematics, Southeast University
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摘要
For the single-layer boundary integral operator$\A_{k,\alpha}=\tfrac12 I+\K_k'+\alpha\cS_k$ of the interior Robin problem for the Helmholtz equation, we study the inverse norm $\norm{\A_{k,\alpha}^{-1}}$ in terms of the spectral family $\Si=\{(k,\alpha):k^2\in\sigma(-\Delta_\alpha)\}$. In the impedance direction we prove a safe-zone bound and a two-sided law $\norm{\A^{-1}}=\Theta(\dist(\alpha,\Si(k))^{-1})$; on the disk the local law becomes an exact equality with a closed-form Bessel constant. The dual frequency-direction estimate has an explicit spectral dependence and an analogous exact local form on the disk. In the positive real parameter domain, the disk has no folds or exceptional points. For spherical shells we distinguish proved normal-form scaling from computation-assisted geometry: numerical resultant continuation resolves nonempty nilpotent arcs, while the proved four-slope law at an exact nilpotent block assigns the half-power $\dist^{-1/2}$ to the eigenvalue channel and to a spectral-to-singular response ratio, with the operator and density norms retaining $\dist^{-1}$. The inverse-norm laws yield continuous residual--stability certificates and a discrete acceptance procedure for a companion density-learning framework. In the reported discrete experiment, all $32$ sampled impedances pass the $10^{-4}$ acceptance-estimate threshold; the remaining discretization and stability-bridge corrections are not interval-enclosed. At $k=0$ the self-adjoint Neumann--Poincar\'e spectral theorem gives the corresponding exact distance law.
Abstract
For the single-layer boundary integral operator$\A_{k,\alpha}=\tfrac12 I+\K_k'+\alpha\cS_k$ of the interior Robin problem for the Helmholtz equation, we study the inverse norm $\norm{\A_{k,\alpha}^{-1}}$ in terms of the spectral family $\Si=\{(k,\alpha):k^2\in\sigma(-\Delta_\alpha)\}$. In the impedance direction we prove a safe-zone bound and a two-sided law $\norm{\A^{-1}}=\Theta(\dist(\alpha,\Si(k))^{-1})$; on the disk the local law becomes an exact equality with a closed-form Bessel constant. The dual frequency-direction estimate has an explicit spectral dependence and an analogous exact local form on the disk. In the positive real parameter domain, the disk has no folds or exceptional points. For spherical shells we distinguish proved normal-form scaling from computation-assisted geometry: numerical resultant continuation resolves nonempty nilpotent arcs, while the proved four-slope law at an exact nilpotent block assigns the half-power $\dist^{-1/2}$ to the eigenvalue channel and to a spectral-to-singular response ratio, with the operator and density norms retaining $\dist^{-1}$. The inverse-norm laws yield continuous residual--stability certificates and a discrete acceptance procedure for a companion density-learning framework. In the reported discrete experiment, all $32$ sampled impedances pass the $10^{-4}$ acceptance-estimate threshold; the remaining discretization and stability-bridge corrections are not interval-enclosed. At $k=0$ the self-adjoint Neumann--Poincar\'e spectral theorem gives the corresponding exact distance law.关键词
Helmholtz equation/Robin boundary condition/boundary integral equation/inverse-norm estimates/exceptional points/a posteriori error certificatesKey words
Helmholtz equation/Robin boundary condition/boundary integral equation/inverse-norm estimates/exceptional points/a posteriori error certificates引用本文复制引用
Ming Yang.Quantitative Inverse-Norm Theory for Robin Boundary Integral Operators[EB/OL].(2026-11-08)[2026-11-11].https://chinaxiv.org/abs/202611.00076.学科分类
数学/物理学
