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Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $
A lower bound for the principal eigenvalue of fully nonlinear elliptic operators
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina. |
$ \lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2 $ |
$ R $ |
$ \Omega\subset {\mathbb R}^n $ |
$ \lambda_{1,p}(\Omega) $ |
$ \lambda_{1,\infty}(\Omega) $ |
$ p $ |
References:
[1] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[2] |
H. Berestycki and L. Rossi,
Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Commun. Pure Appl. Math., 68 (2015), 1014-1065.
doi: 10.1002/cpa.21536. |
[3] |
I. Birindelli and F. Demengel,
First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.
|
[4] |
I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 261-287.
|
[5] |
I. Birindelli and F. Demengel,
Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl., 352 (2009), 822-835.
doi: 10.1016/j.jmaa.2008.11.012. |
[6] |
I. Birindelli and F. Demengel,
Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differ. Equ., 249 (2010), 1089-1110.
doi: 10.1016/j.jde.2010.03.015. |
[7] |
J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for {P}ucci's operators, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
[8] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. N.S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[9] |
G. Crasta and I. Fragalà,
Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal., 133 (2016), 228-249.
doi: 10.1016/j.na.2015.12.007. |
[10] |
P. Juutinen,
Principal eigenvalue of a very badly degenerate operator and applications, J. Differ. Equ., 236 (2007), 532-550.
doi: 10.1016/j.jde.2007.01.020. |
[11] |
P. Juutinen, P. Lindqvist and J. J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
[12] |
B. Kawohl and J. Horák,
On the geometry of the $p$-Laplacian operator, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 799-813.
doi: 10.3934/dcdss.2017040. |
[13] |
B. Kawohl, S. Krömer and J. Kurtz,
Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.
|
[14] |
P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi,
The limit as $p\rightarrow\infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.
doi: 10.1007/s13163-013-0124-4. |
[15] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |
show all references
References:
[1] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[2] |
H. Berestycki and L. Rossi,
Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Commun. Pure Appl. Math., 68 (2015), 1014-1065.
doi: 10.1002/cpa.21536. |
[3] |
I. Birindelli and F. Demengel,
First eigenvalue and maximum principle for fully nonlinear singular operators, Adv. Differ. Equ., 11 (2006), 91-119.
|
[4] |
I. Birindelli and F. Demengel,
Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math. (6), 13 (2004), 261-287.
|
[5] |
I. Birindelli and F. Demengel,
Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl., 352 (2009), 822-835.
doi: 10.1016/j.jmaa.2008.11.012. |
[6] |
I. Birindelli and F. Demengel,
Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differ. Equ., 249 (2010), 1089-1110.
doi: 10.1016/j.jde.2010.03.015. |
[7] |
J. Busca, M. J. Esteban and A. Quaas,
Nonlinear eigenvalues and bifurcation problems for {P}ucci's operators, Ann. Inst. Henri Poincare Anal. Non Lineaire, 22 (2005), 187-206.
doi: 10.1016/j.anihpc.2004.05.004. |
[8] |
M. G. Crandall, H. Ishii and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. N.S., 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[9] |
G. Crasta and I. Fragalà,
Characterization of stadium-like domains via boundary value problems for the infinity Laplacian, Nonlinear Anal., 133 (2016), 228-249.
doi: 10.1016/j.na.2015.12.007. |
[10] |
P. Juutinen,
Principal eigenvalue of a very badly degenerate operator and applications, J. Differ. Equ., 236 (2007), 532-550.
doi: 10.1016/j.jde.2007.01.020. |
[11] |
P. Juutinen, P. Lindqvist and J. J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
[12] |
B. Kawohl and J. Horák,
On the geometry of the $p$-Laplacian operator, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 799-813.
doi: 10.3934/dcdss.2017040. |
[13] |
B. Kawohl, S. Krömer and J. Kurtz,
Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball, Differ. Integral Equ., 27 (2014), 659-670.
|
[14] |
P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi,
The limit as $p\rightarrow\infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.
doi: 10.1007/s13163-013-0124-4. |
[15] |
A. Quaas and B. Sirakov,
Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math., 218 (2008), 105-135.
doi: 10.1016/j.aim.2007.12.002. |




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