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

Stefania Patrizi

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 575-601
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Patrizi, 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
  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.  Zbl0633.35061
  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.  Zbl1150.35047
  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.  Zbl0806.35129
  4. I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully nonlinear homogeneous operators. Comm. Pure Appl. Anal.6 (2007) 335–366.  Zbl1132.35032
  5. 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.  Zbl1205.35087
  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.  Zbl0755.35015
  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).  Zbl0920.49004
  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.  Zbl1114.35072
  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.  Zbl0708.35031
  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.  Zbl1132.35066
  12. P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian. Math. Ann.335 (2006) 819–851.  Zbl1110.35037
  13. P. Juutinen, P. Lindqvist and J.J. Manfredi, The ∞-eigenvalue problem. Arch. Ration. Mech. Anal.148 (1999) 89–105.  Zbl0947.35104
  14. P. Lindqvist, On a nonlinear eigenvalue problem. Report 68, Univ. Jyväskylä, Jyväskylä (1995) 33–54.  Zbl0838.35094
  15. P.L. Lions, Bifurcation and optimal stochastic control. Nonlinear Anal.7 (1983) 177–207.  
  16. S. Patrizi, The Neumann problem for singular fully nonlinear operators. J. Math. Pures Appl.90 (2008) 286–311.  Zbl1185.35087
  17. S. Patrizi, Principal eigenvalues for Isaacs operators with Neumann boundary conditions. NoDEA16 (2009) 79–107.  Zbl1178.35180
  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.  Zbl1206.91002
  19. 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.  Zbl1174.35373
  20. A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear operators. Adv. Math.218 (2008) 105–135.  Zbl1143.35077

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.