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

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

1. 

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

Received  July 2011 Revised  October 2011 Published  June 2012

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.
Citation: Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1137/1018114.  Google Scholar

[2]

H. Amann, Dual semigroups and second order linear elliptic boundaryvalue problems,, Israel J. Math., 45 (1983), 225.  doi: 10.1007/BF02774019.  Google Scholar

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinearindefinite elliptic problems,, Journal of Differential Equations, 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[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.  doi: 10.1006/jdeq.2000.4003.  Google Scholar

[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.  doi: 10.1016/S0362-546X(01)00116-X.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[8]

E. Krahn, Uber eine von Rayleigh formulierte Minimale igenschaft des Kreises,, Math. Ann., 91 (1925), 97.  doi: 10.1007/BF01208645.  Google Scholar

[9]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupledelliptic systems and some applications,, Differential IntegralEquations, 7 (1994), 383.   Google Scholar

[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.  doi: 10.1006/jdeq.1996.0070.  Google Scholar

[11]

J. López-Gómez, "The Strong Maximum Principle,", preprint, (2011).   Google Scholar

[12]

E. M. Stein, "Singular Integrals of Differentiability Propertiesof Functions,", Princeton Univ. Press, (1970).   Google Scholar

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, SIAM Rev., 18 (1976), 620.  doi: 10.1137/1018114.  Google Scholar

[2]

H. Amann, Dual semigroups and second order linear elliptic boundaryvalue problems,, Israel J. Math., 45 (1983), 225.  doi: 10.1007/BF02774019.  Google Scholar

[3]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinearindefinite elliptic problems,, Journal of Differential Equations, 146 (1998), 336.  doi: 10.1006/jdeq.1998.3440.  Google Scholar

[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.  doi: 10.1006/jdeq.2000.4003.  Google Scholar

[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.  doi: 10.1016/S0362-546X(01)00116-X.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jdeq.1996.0071.  Google Scholar

[8]

E. Krahn, Uber eine von Rayleigh formulierte Minimale igenschaft des Kreises,, Math. Ann., 91 (1925), 97.  doi: 10.1007/BF01208645.  Google Scholar

[9]

J. López-Gómez and M. Molina-Meyer, The maximum principle for cooperative weakly coupledelliptic systems and some applications,, Differential IntegralEquations, 7 (1994), 383.   Google Scholar

[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.  doi: 10.1006/jdeq.1996.0070.  Google Scholar

[11]

J. López-Gómez, "The Strong Maximum Principle,", preprint, (2011).   Google Scholar

[12]

E. M. Stein, "Singular Integrals of Differentiability Propertiesof Functions,", Princeton Univ. Press, (1970).   Google Scholar

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