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Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part. (English) Zbl 1091.35046

Summary: An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which are in general nonlinear and have partially Dirichlet and partially Neumann boundary conditions. Its consequences and those of an adapted Hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the shape of the Dirichlet and Neumann boundaries.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] Arrieta, J.M., Neumann eigenvalue problems on exterior perturbations of the domain, J. differential equations, 118, 54-103, (1995) · Zbl 0860.35086
[2] Boggio, T., Sull’equazione del moto vibratorio delle membrane elastiche, Accad. lincei, sci. fis. ser. 5a, 16, 386-393, (1907) · JFM 38.0813.02
[3] Burenkov, V.I.; Davies, E.B., Spectral stability of the Neumann Laplacian, J. differential equations, 186, 2, 485-508, (2002) · Zbl 1042.35035
[4] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II, Interscience Wiley, New York, 1962 (original publication 1937). · Zbl 0788.00012
[5] Courtois, G., Spectrum of manifolds with holes, J. funct. anal., 134, 194-221, (1995) · Zbl 0847.58076
[6] E.B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, MA, 1989. · Zbl 0699.35006
[7] E.B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, MA, 1995. · Zbl 0893.47004
[8] Drábek, P.; Krejčí, P.; Takáč, P., Nonlinear differential equations, (1999), CRC Press Boca Raton, FL
[9] P. Drábek, A. Kufner, F. Nicolosi, Nonlinear Differential Equations, Singular and Degenerate Case, University of West Bohemia, Pilsen, Czech Republic, 1996.
[10] Edmunds, D.E.; Evans, W.D., Spectral theory and differential operators, (1987), Clarendon Press Oxford · Zbl 0628.47017
[11] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002
[12] Evans, W.D.; Harris, D.J., On the approximation numbers of Sobolev embeddings for irregular domains, Quart. J. math. Oxford, 40, 2, 13-42, (1989) · Zbl 0681.46033
[13] Fleckinger, J.; Harrell II, E.M.; Thélin, F.de., Boundary behavior and \(L^q\) estimates for solutions of equations containing the p-Laplacian, Electron. J. differential equation, 1999, 1-19, (1999) · Zbl 0928.35046
[14] Fraenkel, L.E., On regularity of the boundary in the theory of Sobolev spaces, Proc. London math. soc., 39, 3, 385-427, (1979) · Zbl 0406.46026
[15] Hardy, G.H., Note on a theorem of Hilbert, Math. Z., 6, 314-317, (1920) · JFM 47.0207.01
[16] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1959), Cambridge University Press Cambridge, MA, (original publication 1934) · Zbl 0634.26008
[17] E.M. Harrell II, Lecture at the Workshop on Partial Differential Equations and Fractals, Toulouse, France, 1993, unpublished.
[18] Hempel, R.; Seco, L.A.; Simon, B., The essential spectrum of Neumann Laplacians on some bounded singular domains, J. funct. anal., 102, 448-483, (1991) · Zbl 0741.35043
[19] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, vol. 83, American Mathematical Society, Providence, RI, 1994. · Zbl 0812.35001
[20] Transl. Math. USSR IZV (1973) 357-387.
[21] Maz’ja, V.G., Sobolev spaces, (1985), Springer New York, (first published 1981)
[22] McGillivray, I., Capacitary estimates for Dirichlet eigenvalues, J. funct. anal., 139, 244-259, (1996) · Zbl 0896.47017
[23] McGillivray, I., Capacitary asymptotic expansion of the groundstate to second order, Comm. partial differential equations, 23, 2219-2252, (1998) · Zbl 0922.47002
[24] B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics, vol. 219, Boston, Longman,1990. · Zbl 0698.26007
[25] Ozawa, S., Singular variation of domains and eigenvalues of the Laplacian, Duke math. J., 48, 767-778, (1981) · Zbl 0483.35064
[26] Ozawa, S., An asymptotic formula for the eigenvalues of the Laplacian in a domain with a small hole, Proc. Japan acad. ser. A, 58, 5-8, (1982) · Zbl 0516.35015
[27] Ozawa, S., Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains—the Neumann condition, Osaka J. math., 22, 639-655, (1985) · Zbl 0579.35065
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