Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights

Abstract

Let A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ⊂ℝ^k. We consider the variational eigenvalue problem

$$\mathcal{A}u = \lambda r{\text{(}}x{\text{)}}u,{\text{ }}x \in \Omega ,$$

((P))

, with Dirichlet or Neumann boundary conditions; here the “weight” r is a real-valued function on Ω which is allowed to change sign in Ω or to be discontinuous. Such problems occur naturally in the study of many nonlinear elliptic equations. In an earlier work [Trans. Amer. Math. Soc. 295 (1986), pp. 305–324], we have determined the leading term for the asymptotics of the eigenvalues λ of (P). In the present paper, we obtain, under more stringent assumptions, the corresponding remainder estimates. More precisely, let N ^±(λ) be the number of positive (respectively, negative) eigenvalues of (P) less than λ>0 (respectively, greater than λ<0); set r _± = max (±r, 0) and $\Omega _ \pm = {\text{\{ }}x \in \Omega :r{\text{(}}x{\text{)}} \gtrless {\text{0\} }}$. We show that

$$N^ \pm {\text{(}}\lambda {\text{) = }}\mathop \smallint \limits_{\Omega _ \pm } {\text{(}}\lambda r{\text{(}}x{\text{))}}^{\frac{k}{{{\text{2}}m}}} {\text{ }}\mu \prime _\mathcal{A} {\text{(}}x{\text{) }}dx + 0{\text{(}}\left| \lambda \right|^{\frac{{k - 1}}{{{\text{2}}m}} + \delta } {\text{) as }}\lambda \to \pm \infty {\text{,}}$$

, where δ>0 and μ^′ _A (x) is the Browder-Gårding density associated with the principal part of A. How small δ can be chosen depends on the “regularity” of the leading coefficients of A, r _±, and of the boundary of Ω _±. These results seem to be new even for positive weights.