Spectral stability of Dirichlet second order uniformly elliptic operators

Abstract

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type

On the Hadamard Formula for Second Order Systems in Non-Smooth Domains

“…Then there are exactly J m eigenvalues of problem (3) in the interval ((λ m−1 + λ m )/2, (λ m + λ m+1 )/2), for which the following asymptotic formula holds:…”

On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

“…There is a vast literature concerning this problem in the 20th century: Hadamard [22] in 1908, Courant and Hilbert [13] in the German edition of 1937, Polya and Szëgo [43] in 1951, Garabedian and Schiffer [19,20] in 1952-1953, Polya and Schiffer [42] in 1953, Schiffer [46] in 1954 and thereafter [2,18,3,27,39,[36][37][38]40,47,16,44,14,17,21]. For more recent advances, we cite the works [29,11,15,35,6,26,25,30,31,4,8,9,33,10,34,32]. We also mention three interesting works on generic properties of eigenvalues and eigenfunctions due to Uhlenbeck [48,49] and Pereira [41] which are closely related to the issue of this paper.…”

Analytic Dependence of Volume Potentials Corresponding to Parametric Families of Fundamental Solutions

“…So for example, we mention Burenkov and Lamberti [1], Henry [12], Keldysh [16], Maz'ya et al [27], Sokolowski and Zolésio [35], and Ward and Keller [37]. We now briefly outline an application of the results of the previous sections to an operator which appears when dealing with the investigation of the dependence of the solution of a boundary value problem upon perturbation of the coefficients of the differential operator, of the domain, and of the data.…”