# American Institute of Mathematical Sciences

September  2011, 10(5): 1315-1329. doi: 10.3934/cpaa.2011.10.1315

## $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains

 1 Dipartimento di Matematica e Informatica, Università di Salerno, P. Grahamstown, Fisciano, SA I-84084

Received  March 2009 Revised  November 2009 Published  April 2011

In this paper we consider estimates of the Raleigh quotient and in general of the $H^{1,p}$-eigenvalue in quasicylindrical domains. Then we apply the results to obtain, by variational methods, existence and uniqueness of weak solutions of the Dirichlet problem for second-order elliptic equations in divergent form. For such solutions global boundedness estimates have been also established.
Citation: Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65. [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. doi: ISBN:0120441500.  Google Scholar [2] H. Brezis, "Analyse fonctionnelle. (French) [Functional analysis] Théorie et applications. [Theory and applications]," Collection Mathématiques Appliquées pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983. doi: ISBN:9782225771989.  Google Scholar [3] X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hlder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.  Google Scholar [4] V. Cafagna and A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains, C. R. Math. Acad. Sci. Paris, 334 (2002), 359-363. doi: 10.1016/S1631-073X(02)02267-7.  Google Scholar [5] I. Capuzzo Dolcetta and A. Vitolo, On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains, Matematiche (Catania), 62 (2007), 69-91. doi: ISSN 0373-3505; ISSN 2037-5298.  Google Scholar [6] I. Ekeland and R. Temam, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. "Convex Analysis and Variational Problems," North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. doi: ISBN:0898714508.  Google Scholar [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer Verlag, Berlin, 2001. doi: ISBN:3540411607.  Google Scholar [8] W. K. Hayman, Some bounds for principal frequency,, Appl. Anal., 7 (): 247.  doi: 10.1080/00036817808839195.  Google Scholar [9] B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667. doi: ISSN:0010-2628.  Google Scholar [10] E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448. doi: 10.1007/BF01394245.  Google Scholar [11] V. Maz'ya and M. A. Shubin, Can one see the fundamental frequency of a drum?, Lett. Math. Phys., 74 (2005), 135-151. doi: ISSN:0377-9017.  Google Scholar [12] R. Osserman, A note on Hayman's theorem on the bass note of a drum, Comment. Math. Helv., 52 (1977), 545-555. doi: 10.1007/BF02567388.  Google Scholar [13] M. Transirico, M. Troisi and A. Vitolo, Spaces of Morrey type and elliptic equations in divergence form on unbounded domains, Boll. Un. Mat. Ital. B (7), 9 (1995), 153-174. doi: ISSN:0392-4041.  Google Scholar [14] A. Vitolo, A note on the maximum principle for complete second-order elliptic operators in general domains, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1955-1966. doi: 10.1007/s10114-007-0976-y.  Google Scholar

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65. [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. doi: ISBN:0120441500.  Google Scholar [2] H. Brezis, "Analyse fonctionnelle. (French) [Functional analysis] Théorie et applications. [Theory and applications]," Collection Mathématiques Appliquées pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983. doi: ISBN:9782225771989.  Google Scholar [3] X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hlder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.  Google Scholar [4] V. Cafagna and A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains, C. R. Math. Acad. Sci. Paris, 334 (2002), 359-363. doi: 10.1016/S1631-073X(02)02267-7.  Google Scholar [5] I. Capuzzo Dolcetta and A. Vitolo, On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains, Matematiche (Catania), 62 (2007), 69-91. doi: ISSN 0373-3505; ISSN 2037-5298.  Google Scholar [6] I. Ekeland and R. Temam, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. "Convex Analysis and Variational Problems," North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. doi: ISBN:0898714508.  Google Scholar [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer Verlag, Berlin, 2001. doi: ISBN:3540411607.  Google Scholar [8] W. K. Hayman, Some bounds for principal frequency,, Appl. Anal., 7 (): 247.  doi: 10.1080/00036817808839195.  Google Scholar [9] B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667. doi: ISSN:0010-2628.  Google Scholar [10] E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448. doi: 10.1007/BF01394245.  Google Scholar [11] V. Maz'ya and M. A. Shubin, Can one see the fundamental frequency of a drum?, Lett. Math. Phys., 74 (2005), 135-151. doi: ISSN:0377-9017.  Google Scholar [12] R. Osserman, A note on Hayman's theorem on the bass note of a drum, Comment. Math. Helv., 52 (1977), 545-555. doi: 10.1007/BF02567388.  Google Scholar [13] M. Transirico, M. Troisi and A. Vitolo, Spaces of Morrey type and elliptic equations in divergence form on unbounded domains, Boll. Un. Mat. Ital. B (7), 9 (1995), 153-174. doi: ISSN:0392-4041.  Google Scholar [14] A. Vitolo, A note on the maximum principle for complete second-order elliptic operators in general domains, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1955-1966. doi: 10.1007/s10114-007-0976-y.  Google Scholar
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