The principal eigenvalue of the -laplacian with the Neumann boundary condition
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 575-601.

We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

DOI : 10.1051/cocv/2010019
Classification : 35J25, 35D40, 35P30, 35J60
Mots clés : ∞-laplacian, Neumann boundary condition, principal eigenvalue, viscosity solutions
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     title = {The principal eigenvalue of the $\infty $-laplacian with the {Neumann} boundary condition},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {575--601},
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Patrizi, Stefania. The principal eigenvalue of the $\infty $-laplacian with the Neumann boundary condition. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 575-601. doi : 10.1051/cocv/2010019. http://www.numdam.org/articles/10.1051/cocv/2010019/

[1] A. Anane, Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 752-728. | MR | Zbl

[2] G. Aronsson, M.G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. (N.S.) 41 (2004) 439-505. | MR | Zbl

[3] H. Berestycki, L. Nirenberg and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domain. Comm. Pure Appl. Math. 47 (1994) 47-92. | MR | Zbl

[4] I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Comm. Pure Appl. Anal. 6 (2007) 335-366. | MR | Zbl

[5] J. Busca, M.J. Esteban, A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 187-206. | Numdam | MR | Zbl

[6] M.C. 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. | MR | Zbl

[7] L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137. American Mathematical Society (1999). | MR | Zbl

[8] J. Garcia-Azorero, J.J. Manfredi, I. Peral and J.D. Rossi, Steklov eigenvalues for the ∞-Laplacian. Rend. Lincei Mat. Appl. 17 (2006) 199-210. | MR | Zbl

[9] H. Ishii and P.L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Diff. Equ. 83 (1990) 26-78. | MR | Zbl

[10] H. Ishii and Y. Yoshimura, Demi-eigenvalues for uniformly elliptic Isaacs operators. Preprint.

[11] P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications. J. Diff. Equ. 236 (2007) 532-550. | MR | Zbl

[12] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian. Math. Ann. 335 (2006) 819-851. | MR | Zbl

[13] P. Juutinen, P. Lindqvist and J.J. Manfredi, The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148 (1999) 89-105. | MR | Zbl

[14] P. Lindqvist, On a nonlinear eigenvalue problem. Report 68, Univ. Jyväskylä, Jyväskylä (1995) 33-54. | MR | Zbl

[15] P.L. Lions, Bifurcation and optimal stochastic control. Nonlinear Anal. 7 (1983) 177-207. | MR

[16] S. Patrizi, The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl. 90 (2008) 286-311. | MR | Zbl

[17] S. Patrizi, Principal eigenvalues for Isaacs operators with Neumann boundary conditions. NoDEA 16 (2009) 79-107. | MR | Zbl

[18] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc. 22 (2009) 167-210. | MR | Zbl

[19] A. Quaas, Existence of positive solutions to a “semilinear” equation involving the Pucci's operators in a convex domain. Diff. Integral Equations 17 (2004) 481-494. | Zbl

[20] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Adv. Math. 218 (2008) 105-135. | MR | Zbl

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