Properties of the Principal Eigenvalues of a General Class of Non-classical Mixed Boundary Value Problems

Abstract

In this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations 5, 999-1030. In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations 138, 86-132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations 146, 336-374).

Maximum Principles and Principal Eigenvalues

“…This corollary is Theorem 4.1 in [21]. A related result can be found in [20] for the Dirichlet problem for irreducible cooperative systems.…”

“…) by López-Gómez [52] for Dirichlet and by Cano-Casanova and López-Gómez [21] for general boundary conditions (also see [37] and, for the Dirichlet problem in a weak setting, [24] has also been shown in [56], given the much more restrictive assumption that and`2…”

“…This theorem, which we derive in Section 18 from Theorem 20, is -except for regularity assumptions on the Dirichlet boundary -a generalization of the continuity result in [21], where the Robin boundary and the coefficients are kept fixed and the scalar case is considered only. Of course, in the case of Dirichlet boundary conditions for a scalar equation much more general results can be obtained as has been shown by Daners [28], [29] and others (see the remarks following Theorem 18).…”

Positive solutions for some indefinite nonlinear eigenvalue elliptic problems with Robin boundary conditions

“…Now, we can show that M δ is a sequence of bounded and regular domains converging to M 0 from the exterior in the sense of [8]. So, by Theorem 7.1 in [8], we conclude that…”

“…Thus, by continuity of α 1 (λ) and ϕ 1 with respect to λ (see for instance [8]), there exists δ > 0 such that…”