The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition
ESAIM: Control, Optimisation and Calculus of Variations (2011)
- Volume: 17, Issue: 2, page 575-601
- ISSN: 1292-8119
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topPatrizi, Stefania. "The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 575-601. <http://eudml.org/doc/197358>.
@article{Patrizi2011,
abstract = {
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.
},
author = {Patrizi, Stefania},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {∞-Laplacian; Neumann boundary condition; principal eigenvalue; viscosity solutions; -Laplacian},
language = {eng},
month = {5},
number = {2},
pages = {575-601},
publisher = {EDP Sciences},
title = {The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition},
url = {http://eudml.org/doc/197358},
volume = {17},
year = {2011},
}
TY - JOUR
AU - Patrizi, Stefania
TI - The principal eigenvalue of the ∞-Laplacian with the Neumann boundary condition
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 575
EP - 601
AB -
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.
LA - eng
KW - ∞-Laplacian; Neumann boundary condition; principal eigenvalue; viscosity solutions; -Laplacian
UR - http://eudml.org/doc/197358
ER -
References
top- 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.
- 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.
- 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.
- I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Comm. Pure Appl. Anal.6 (2007) 335–366.
- J. Busca, M.J. Esteban, A. Quaas, Nonlinear eigenvalues and bifurcation problems for Pucci's operators. Ann. Inst. H. Poincaré Anal. Non Linéaire22 (2005) 187–206.
- 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.
- 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).
- 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.
- H. Ishii and P.L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Diff. Equ.83 (1990) 26–78.
- H. Ishii and Y. Yoshimura, Demi-eigenvalues for uniformly elliptic Isaacs operators. Preprint.
- P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications. J. Diff. Equ.236 (2007) 532–550.
- P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian. Math. Ann.335 (2006) 819–851.
- P. Juutinen, P. Lindqvist and J.J. Manfredi, The ∞-eigenvalue problem. Arch. Ration. Mech. Anal.148 (1999) 89–105.
- P. Lindqvist, On a nonlinear eigenvalue problem. Report 68, Univ. Jyväskylä, Jyväskylä (1995) 33–54.
- P.L. Lions, Bifurcation and optimal stochastic control. Nonlinear Anal.7 (1983) 177–207.
- S. Patrizi, The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl.90 (2008) 286–311.
- S. Patrizi, Principal eigenvalues for Isaacs operators with Neumann boundary conditions. NoDEA16 (2009) 79–107.
- Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc.22 (2009) 167–210.
- A. Quaas, Existence of positive solutions to a “semilinear” equation involving the Pucci's operators in a convex domain. Diff. Integral Equations17 (2004) 481–494.
- A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Adv. Math.218 (2008) 105–135.
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