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Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems. (English) Zbl 1086.35073
Summary: 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 [Commun. Partial Differ. Equations 5, 999–1030 (1980; Zbl 0477.35075)]. 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 [J. Differ. Equations 138, 86–132 (1997; Zbl 0886.35063)]. 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 [J. Differ. Equations 146, 336–374 (1998; Zbl 0909.35044)].

35P15 Estimates of eigenvalues in context of PDEs
35P05 General topics in linear spectral theory for PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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