American Institute of Mathematical Sciences

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

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

Antonio Vitolo 1,

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.
Keywords: eigenvalues, Elliptic equations, Dirichlet problem, pointwise estimates..
Mathematics Subject Classification: Primary: 35P15, 35J25; Secondary: 35B45.
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

show all references

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

[1]

Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure & Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024
[2]

Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209
[3]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[4]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[5]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709
[6]

Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141
[7]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961
[8]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
[9]

E. N. Dancer, Danielle Hilhorst, Shusen Yan. Peak solutions for the Dirichlet problem of an elliptic system. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 731-761. doi: 10.3934/dcds.2009.24.731
[10]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[11]

Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure & Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119
[12]

Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139
[13]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4461-4476. doi: 10.3934/dcds.2021043
[14]

Farman Mamedov, Sara Monsurrò, Maria Transirico. Potential estimates and applications to elliptic equations. Conference Publications, 2015, 2015 (special) : 793-800. doi: 10.3934/proc.2015.0793
[15]

Niklas Behringer. Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions. Mathematical Control & Related Fields, 2021, 11 (2) : 313-328. doi: 10.3934/mcrf.2020038
[16]

Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010
[17]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101
[18]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264
[19]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[20]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences