Asymptotic distribution of eigenvalues and eigenfunctions for general linear elliptic boundary value problems
Bùi An Tôn · Transactions of the American Mathematical Society · 1966
The asymptotic distribution of eigenvalues and eigenfunctions of elliptic operators has been studied extensively by Weyl, Courant, Carleman, Pleijel and others.During the last decade, Girding [9] and Browder solved the problem for an elliptic operator with infinitely differentiable coefficients and null Dirichlet boundary conditions.It is the purpose of this paper to consider the problem for a general class of elliptic boundary value problems investigated during the last few years by Agmon-Douglis-Nirenberg [2], Browder [3], [6] and Schechter [14].Let (/I,y) with y = {By,---,Bm} be a uniformly regularly elliptic boundary value problem on S. It is assumed that A is positively strongly elliptic and 04, y) is formally positive.If Ay is the realization of A as an operator on L2(S) with null boundary conditions y, then the following results are obtained: (1) When Ay is self-adjoint: N(t) = £ l~(27i)-"tn/2m f f dÇdx así-+ 00.Xj¿t J S J a(x,Ç) o + ir2p^ for x in S as t -> + oo (4mp > n + | a | + | ß |).