×

zbMATH — the first resource for mathematics

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)].

MSC:
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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, D.R.; Hedberg, L.I., Function spaces and potential theory, Grundlehren der mathematischen wissenschaften, 314, (1996), Springer-Verlag Berlin
[2] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[3] Amann, H., Dual semigroups and second order linear elliptic boundary value problems, Israel J. math., 45, 225-254, (1983) · Zbl 0535.35017
[4] Amann, H.; López-Gómez, J., A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. differential equations, 146, 336-374, (1998) · Zbl 0909.35044
[5] Arrieta, J.M., Rate of eigenvalues on a dumbbell domain, simple eigenvalues case, Trans. amer. math. soc., 347, 3503-3531, (1995) · Zbl 0856.35095
[6] Arrieta, J.M.; Hale, J.K.; Han, Q., Eigenvalue problems for nonsmoothly perturbed domains, J. differential equations, 91, 24-52, (1991) · Zbl 0736.35073
[7] Babuska, I., Stability of the domain with respect to the fundamental problems in the theory of partial differential equations, mainly in connection with the theory of elasticity, I, II, Czechoslovak. math. J., 11, 76-105, (1961)
[8] Babuska, I.; Vyborny, R., Continuous dependence of eigenvalues on the domain, Czechoslovak. math. J., 15, 169-178, (1965) · Zbl 0137.32302
[9] Berestycki, H.; Nirenberg, L.; Varadhan, S.R.S., The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. pure appl. math., 47, 47-92, (1994) · Zbl 0806.35129
[10] Brezis, H., Analyse fonctionnelle, (1983), Masson Paris · Zbl 0511.46001
[11] Courant, R.; Hilbert, D., Methods of mathematical physics, (1962), Wiley-Interscience New York · Zbl 0729.35001
[12] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. rational. mech. anal., 67, 161-180, (1973) · Zbl 0275.47044
[13] Dancer, E.N., Some remarks on classical problems and fine properties of Sobolev spaces, Differential integral equations, 9, 437-446, (1996) · Zbl 0853.35011
[14] Dancer, E.N.; Daners, D., Domain perturbation for elliptic equations subject to Robin boundary conditions, J. differential equations, 138, 86-132, (1997) · Zbl 0886.35063
[15] E. N. Dancer, and, J. López-Gómez, Semiclassical analysis of general second order elliptic operators on bounded domains, Trans. Amer. Math. Soc, in press.
[16] Faber, C., Beweis das unter Allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt, Sitzungsber. bayer. akad. wiss. math. phys., 169-172, (1923) · JFM 49.0342.03
[17] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of the second order, (1983), Springer-Verlag Berlin · Zbl 0691.35001
[18] Guzmán, M.; Rubio, B., Integración: teorı́a y Técnicas, (1979), Alhambra Madrid
[19] Hale, J.K.; Vegas, J.M., A nonlinear parabolic equation with varying domain, Arch. rational mech. anal., 86, 99-123, (1984) · Zbl 0569.35048
[20] Hedberg, L.I., Approximation by harmonic functions, and stability of the Dirichlet problem, Exposition. math, 11, 193-259, (1993) · Zbl 0781.31001
[21] Hess, P., Periodic-parabolic boundary value problems and positivity, Pitman res. notes math. ser., 247, (1991), Longman Harlow
[22] Hess, P.; Kato, T., On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. partial differential equations, 5, 999-1030, (1980) · Zbl 0477.35075
[23] Jimbo, S., The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary condition, J. differential equations, 77, 322-350, (1989) · Zbl 0703.35138
[24] Jimbo, S., Perturbation formula of eigenvalues in a singularly perturbed domain, J. math. soc. Japan, 45, 339-356, (1993) · Zbl 0785.35069
[25] Kato, T., Superconvexity of the spectral radius and convexity of the spectral bound and type, Math. Z., 180, 265-273, (1982) · Zbl 0471.46012
[26] Kato, T., Perturbation theory for linear operators, Classics in mathematics, (1995), Springer-Verlag Berlin
[27] Krahn, E., Über eine von Rayleigh formulierte minimaleigenschaft des kreises, Math. ann., 91, 97-100, (1925) · JFM 51.0356.05
[28] López-Gómez, J., On linear weighted boundary value problems, (), 188-203 · Zbl 0815.35014
[29] López-Gómez, J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. differential equations, 127, 263-294, (1996) · Zbl 0853.35078
[30] López-Gómez, J.; Molina-Meyer, M., The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential integral equations, 7, 383-398, (1994) · Zbl 0827.35019
[31] Manes, A.; Micheletti, A.M., Un’estensione Della teoria variazonale classica degli autovalori per operatori ellittici del secondo ordine, Boll. un. mat. ital., 7, 285-301, (1973) · Zbl 0275.49042
[32] Pinsky, R.G., Positive harmonic functions and diffusion, an integrated analytic and probabilistic approach, Cambridge studies in advanced mathematics, 45, (1995), Cambridge Univ. Press Cambridge
[33] Protter, M.; Weinberger, H., On the spectrum of general second order operators, Boll. amer. math. soc., 72, 251-255, (1966) · Zbl 0141.09901
[34] Schaefer, H.H., Topological vector spaces, (1971), Springer-Verlag New York · Zbl 0217.16002
[35] Simon, B., Semiclassical analysis of low lying eigenvalues. I. non-degenerate minima: asymptotic expansions, Ann. inst. H. Poincaré A, 38, 12-37, (1983)
[36] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
[37] Ward, M.J.; Keller, J.B., Strong localized perturbations of eigenvalue problems, SIAM J. appl. math., 53, 770-798, (1993) · Zbl 0778.35081
[38] Ward, M.J.; Henshaw, D.; Keller, J.B., Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAM J. appl. math., 53, 799-828, (1993) · Zbl 0778.35082
[39] J. Wloka, Partial Differential Equations, Cambridge Univ. Press, Cambridge, UK, 1987. · Zbl 0623.35006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.