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The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition

Published online by Cambridge University Press:  10 May 2010

Stefania Patrizi
Stefania Patrizi*
Affiliation:
SAPIENZA Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, 00185 Roma, Italy. [email protected]
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Abstract

We prove the existence of a principal eigenvalue associated to the∞-Laplacian plus lower order terms and the Neumann boundarycondition in a bounded smooth domain. As an application we getuniqueness and existence results for the Neumann problem and adecay estimate for viscosity solutions of the Neumann evolutionproblem.

Keywords

∞-LaplacianNeumann boundary conditionprincipal eigenvalueviscosity solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 575 - 601
Copyright
© EDP Sciences, SMAI, 2010

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References

Anane, A., 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.
Aronsson, G., Crandall, M.G. and Juutinen, P., A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. (N.S.) 41 (2004) 439–505. CrossRef
Berestycki, H., Nirenberg, L. and Varadhan, S.R.S., The principal eigenvalue and maximum principle for second order elliptic operators in general domain. Comm. Pure Appl. Math. 47 (1994) 47–92. CrossRef
Birindelli, I. and Demengel, F., Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Comm. Pure Appl. Anal. 6 (2007) 335–366.
Busca, J., Esteban, M.J., Quaas, A., Nonlinear eigenvalues and bifurcation problems for Pucci's operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 187–206. CrossRef
Crandall, M.C., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. CrossRef
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).
Garcia-Azorero, J., Manfredi, J.J., Peral, I. and Rossi, J.D., Steklov eigenvalues for the ∞-Laplacian. Rend. Lincei Mat. Appl. 17 (2006) 199–210.
Ishii, H. and Lions, P.L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Diff. Equ. 83 (1990) 26–78. CrossRef
H. Ishii and Y. Yoshimura, Demi-eigenvalues for uniformly elliptic Isaacs operators. Preprint.
Juutinen, P., Principal eigenvalue of a very badly degenerate operator and applications. J. Diff. Equ. 236 (2007) 532–550. CrossRef
Juutinen, P. and Kawohl, B., On the evolution governed by the infinity Laplacian. Math. Ann. 335 (2006) 819–851. CrossRef
Juutinen, P., Lindqvist, P. and Manfredi, J.J., The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148 (1999) 89–105. CrossRef
P. Lindqvist, On a nonlinear eigenvalue problem. Report 68, Univ. Jyväskylä, Jyväskylä (1995) 33–54.
Lions, P.L., Bifurcation and optimal stochastic control. Nonlinear Anal. 7 (1983) 177–207. CrossRef
Patrizi, S., The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl. 90 (2008) 286–311. CrossRef
Patrizi, S., Principal eigenvalues for Isaacs operators with Neumann boundary conditions. NoDEA 16 (2009) 79–107. CrossRef
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D., Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc. 22 (2009) 167–210. CrossRef
Quaas, A., Existence of positive solutions to a “semilinear” equation involving the Pucci's operators in a convex domain. Diff. Integral Equations 17 (2004) 481–494.
Quaas, A. and Sirakov, B., Principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Adv. Math. 218 (2008) 105–135. CrossRef