Abstract
Let A be a positive self-adjoint elliptic operator of order 2m on a bounded open set Ω ⊂ℝk. We consider the variational eigenvalue problem
, 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
, 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.
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Communicated by J. B. McLeod
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Fleckinger, J., Lapidus, M.L. Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights. Arch. Rational Mech. Anal. 98, 329–356 (1987). https://doi.org/10.1007/BF00276913
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DOI: https://doi.org/10.1007/BF00276913