A minimax formula for principal eigenvalues and application to an antimaximum principle

Abstract

A minimax formula for the principal eigenvalue of a nonselfadjoint elliptic problem was established in [17]. In this paper we extend this formula to the case where an indefinite weight is present. An application is given to the study of the uniformity of the antimaximum principle. (2000): 35J20, 35P15 Mathematics Subject Classification

The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator

“…This last formula is the exact generalization of the formula proved by M. Donsker and S. Varadhan [14] and by C. J. Holland [26] in bounded domains with Dirichlet or Neumann boudary conditions. This formula has been generalized to the principal eigenvalues of an elliptic operator with undefinite weight by T. Godoy, J.-P. Gossez and S. Paczka [18] and to parabolic operators by T. Godoy, U. Kaufmann and S. Paczka [19].…”

On the asymptotic behavior of the principal eigenvalues of some elliptic problems

“…This guarantees that the infimum over u in the minimax formula is achieved. Of course, references should be made to Lemmas 3.2 and 3.4 from [8] instead of Lemmas 3.2 and 3.3 from [9]. In the proof of (ii) from Proposition 4.4, one should also replace a 0 ≥ 0 by a 0 ≥ 0 and either a 0 ≡ 0 or b 0 ≡ 0.…”

“…e.g. [8,10,17]). In case of existence we will again denote by λ α the largest principal eigenvalue of (6.1) α .…”

“…The following lemma (which does not depend on assumption (3.1)) collects some properties of this function. References include [2,10,13,16]; the proof of most of the properties below can for instance be adapted from the proof of Lemma 2.5 in [8].…”

“…Now, a simple calculation based on the idea of completing a square gives that for any x ∈ and ξ ∈ R N , (3.11) in [8]). We apply the inequality ≤ provided by the above relation with ξ = −∇u * / (u * + ε) to derive from (2.7) that for any measurable function η:…”