Coercivity of elliptic mixed boundary value problems in annulus of \mathbb{R}^N

Previous Article
Dynamics of a reaction-diffusion-advection model for two competing species
DCDS Home
This Issue
Next Article
Boundary estimates for solutions of weighted semilinear elliptic equations

November 2012, 32(11): 3819-3839. doi: 10.3934/dcds.2012.32.3819

Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$

Santiago Cano-Casanova ^1,

1.	Departamento de Matemática Aplicada y Computación, Escuela Técnica Superior de Ingeniería - ICAI, Universidad Ponticia Comillas, Alberto Aguilera, 25, 28015-Madrid, Spain

Received July 2011 Revised October 2011 Published June 2012

Abstract
Related Papers

In this paper is proved that the Strong Maximum Principle is satisfied for a wide class of linear elliptic boundary value problems of mixed type in an annulus of $\mathbb{R}^N$, $N\geq 1$, provided it is thin enough. The coercive character of these boundary value problems is obtained thanks to the characterization of the Strong Maximum Principle found in [3], proving that the principal eigenvalue associated to each boundary value problem may be as large as we wish, independently of the weight on the boundary, by taking the annulus thin enough.

Keywords: principal eigenvalues, Maximum principle, coercivity of elliptic problems and mixed boundary conditions..

Mathematics Subject Classification: Primary: 34B18, 35P15; Secondary: 34B15, 34B0.

Citation: Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819

References:

[1]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.
[2]	H. Amann, Dual semigroups and second order linear elliptic boundaryvalue problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinearindefinite elliptic problems, Journal of Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general classof non-classical mixed boundary value problems, Journal ofDifferential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.
[5]	S. Cano-Casanova, Existence and structure of the set ofpositive solutions of a general class of sublinear elliptic non-classical mixedboundary value problems, Nonlinear Analysis, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.
[6]	C. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicherSpannung die Kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math. Phys., (1923), 169-172.
[7]	J. M. Fraile, P. Koch Medina, J. López-Gómez and S.Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, Journal of Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.
[8]	E. Krahn, Uber eine von Rayleigh formulierte Minimale igenschaft des Kreises, Math. Ann., 91 (1925), 97-100. doi: 10.1007/BF01208645.
[9]	J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupledelliptic systems and some applications, Differential IntegralEquations, 7 (1994), 383-398.
[10]	J. López-Gómez, The maximum principle and the existence of principaleigenvalues for some linear weighted boundary value problems, Journal of Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.
[11]	J. López-Gómez, "The Strong Maximum Principle," preprint, 2011.
[12]	E. M. Stein, "Singular Integrals of Differentiability Propertiesof Functions," Princeton Univ. Press, Princeton, NJ, 1970.

show all references

References:

[1]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114.
[2]	H. Amann, Dual semigroups and second order linear elliptic boundaryvalue problems, Israel J. Math., 45 (1983), 225-254. doi: 10.1007/BF02774019.
[3]	H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinearindefinite elliptic problems, Journal of Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440.
[4]	S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general classof non-classical mixed boundary value problems, Journal ofDifferential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.
[5]	S. Cano-Casanova, Existence and structure of the set ofpositive solutions of a general class of sublinear elliptic non-classical mixedboundary value problems, Nonlinear Analysis, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X.
[6]	C. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicherSpannung die Kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math. Phys., (1923), 169-172.
[7]	J. M. Fraile, P. Koch Medina, J. López-Gómez and S.Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, Journal of Differential Equations, 127 (1996), 295-319. doi: 10.1006/jdeq.1996.0071.
[8]	E. Krahn, Uber eine von Rayleigh formulierte Minimale igenschaft des Kreises, Math. Ann., 91 (1925), 97-100. doi: 10.1007/BF01208645.
[9]	J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupledelliptic systems and some applications, Differential IntegralEquations, 7 (1994), 383-398.
[10]	J. López-Gómez, The maximum principle and the existence of principaleigenvalues for some linear weighted boundary value problems, Journal of Differential Equations, 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.
[11]	J. López-Gómez, "The Strong Maximum Principle," preprint, 2011.
[12]	E. M. Stein, "Singular Integrals of Differentiability Propertiesof Functions," Princeton Univ. Press, Princeton, NJ, 1970.

[1]	Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225
[2]	Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[3]	Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
[4]	Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3461-3471. doi: 10.3934/dcdss.2020239
[5]	Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[6]	Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
[7]	Alassane Niang. Boundary regularity for a degenerate elliptic equation with mixed boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (1) : 107-128. doi: 10.3934/cpaa.2019007
[8]	Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024
[9]	Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061
[10]	Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[11]	José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[12]	Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure and Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254
[13]	J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[14]	Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034
[15]	Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3495-3502. doi: 10.3934/dcdss.2020248
[16]	Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[17]	Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[18]	Vesselin Petkov. Location of eigenvalues for the wave equation with dissipative boundary conditions. Inverse Problems and Imaging, 2016, 10 (4) : 1111-1139. doi: 10.3934/ipi.2016034
[19]	Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[20]	Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Santiago Cano-Casanova
on Google Scholar
Santiago Cano-Casanova

[Back to Top]