Spectral Bounds and Stability Estimates for Dirichlet and Neumann Laplacian with Lebesque Classes

Abstract

We show the approximation of ground state eigenvalues and eigenfunctions, with properties of convergence, of Dirichlet Laplacians and structures. The extension property and boundary regularity for the poisson equation of Refenberg-flat domains are considered. We study the non-selfadiont spectral problems for liner pencils of ordinary differential operators with λ- linear boundary conditions and the characterizations of the spectral decomposition of the non self-adjont block operater matrices. We apply the integration of positive constructible functions against Euler characteristic and dimension with the loci of integrability, zero loci, and stability under integration for the preparation of the constructible functions on Euclidean space with Lebesque measure and classes. The quantitative and spectral stability for the first Dirichlet eigenvalue and Neuman Laplacian in Refenberg-flat and rough domains in Euclidean space are shown. We discuss the class of non-selfadjoint quadratic matrix operator pencils arising in elasticity theory and the spectral sharp bounds and basis results for non selfadjoint pencils with an application to Hagen-poiseuille flow.