The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞

Marino Belloni; Bernd Kawohl

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 1, page 28-52
  • ISSN: 1292-8119

Abstract

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We consider the pseudo-p-Laplacian, an anisotropic version of the p-Laplacian operator for p 2 . We study relevant properties of its first eigenfunction for finite p and the limit problem as p → ∞.

How to cite

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Belloni, Marino, and Kawohl, Bernd. "The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 28-52. <http://eudml.org/doc/90720>.

@article{Belloni2010,
abstract = { We consider the pseudo-p-Laplacian, an anisotropic version of the p-Laplacian operator for $p\not=2$. We study relevant properties of its first eigenfunction for finite p and the limit problem as p → ∞. },
author = {Belloni, Marino, Kawohl, Bernd},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Eigenvalue; anisotropic; pseudo-Laplace; viscosity solution; minimal Lipschitz extension; concavity; symmetry; convex rearrangement; pseudo--Laplacian operator; symmetry},
language = {eng},
month = {3},
number = {1},
pages = {28-52},
publisher = {EDP Sciences},
title = {The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞},
url = {http://eudml.org/doc/90720},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Belloni, Marino
AU - Kawohl, Bernd
TI - The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 28
EP - 52
AB - We consider the pseudo-p-Laplacian, an anisotropic version of the p-Laplacian operator for $p\not=2$. We study relevant properties of its first eigenfunction for finite p and the limit problem as p → ∞.
LA - eng
KW - Eigenvalue; anisotropic; pseudo-Laplace; viscosity solution; minimal Lipschitz extension; concavity; symmetry; convex rearrangement; pseudo--Laplacian operator; symmetry
UR - http://eudml.org/doc/90720
ER -

References

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  1. W. Allegretto and Yin Xi Huang, A Picone's identity for the p-Laplacian and applications. Nonlin. Anal. TMA32 (1998) 819-830.  Zbl0930.35053
  2. A. Alvino, V. Ferone, G. Trombetti and P.L. Lions, Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997) 275-293.  Zbl0877.35040
  3. 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) 725-728.  Zbl0633.35061
  4. A. Anane, A. Benazzi and O. Chakrone, Sur le spectre d'un opérateur quasilininéaire elliptique "dégénéré". Proyecciones19 (2000) 227-248.  
  5. G. Aronsson, Extension of functions satisfying Lipschitz conditions. Ark. Math.6 (1967) 551-561.  Zbl0158.05001
  6. G. Aronsson, On the partial differential equation u x 2 u x x + 2 u x u y u x y + u y 2 u y y = 0 . Ark. Math.7 (1968) 395-425.  
  7. G. Barles, Remarks on uniqueness results of the first eigenvalue of the p-Laplacian. Ann. Fac. Sci. Toulouse9 (1988) 65-75.  Zbl0621.35068
  8. G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Partial Differential Equations26 (2001) 2323-2337.  Zbl0997.35023
  9. M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator. Manuscripta Math.109 (2002) 229-231.  Zbl1100.35032
  10. T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of Δpup = ƒ and related extremal problems. Rend. Sem. Mat., Fasciolo Speciale Nonlinear PDE's. Univ. Torino (1989) 15-68.  
  11. T. Bhattacharya, An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions. Electron. J. Differential Equations2001 (2001) 1-8.  Zbl0966.35052
  12. H. Brezis and L.Oswald, Remarks on sublinear problems. Nonlinear Anal.10 (1986) 55-64.  Zbl0593.35045
  13. M.G. Crandall, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differential Equations13 (2001) 123-139.  Zbl0996.49019
  14. 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.  Zbl0755.35015
  15. Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom.33 (1991) 749-786.  Zbl0696.35087
  16. J.I. Diaz and J.E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math.305 (1987) 521-524.  Zbl0656.35039
  17. E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA7 (1983) 827-850.  Zbl0539.35027
  18. A. Elbert, A half-linear second order differential equation. Qualitative theory of differential equations, (Szeged 1979). Colloq. Math. Soc. János Bolyai30 (1981) 153-180.  
  19. N. Fukagai, M. Ito and K. Narukawa, Limit as p → ∞ of p-Laplace eigenvalue problems and L∞ inequality of the Poincaré type. Differ. Integral Equations 12 (1999) 183-206.  Zbl1064.35512
  20. M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math.148 (1982) 31-46.  Zbl0494.49031
  21. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of second Order. Springer Verlag, Berlin-Heidelberg-New York (1977).  Zbl0361.35003
  22. T. Ishibashi and S. Koike, On fully nonlinear pdes derived from variational problems of Lp-norms. SIAM J. Math. Anal.33 (2001) 545-569.  Zbl1030.35088
  23. U. Janfalk, Behaviour in the limit, as p → ∞, of minimizers of functionals involving p-Dirichlet integrals. SIAM J. Math. Anal.27 (1996) 341-360.  Zbl0853.35028
  24. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal.123 (1993) 51-74.  Zbl0789.35008
  25. P. Juutinen, Personal Communications.  
  26. P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem. Arch. Rational Mech. Anal.148 (1999) 89-105.  Zbl0947.35104
  27. B. Kawohl, Rearrangements and convexity of level sets in PDE. Springer, Lecture Notes in Math. 1150 (1985).  
  28. B. Kawohl, A family of torsional creep problems. J. Reine Angew. Math.410 (1990) 1-22.  
  29. B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete Contin. Dynam. Systems6 (2000) 683-690.  Zbl1157.35342
  30. B. Kawohl and N. Kutev, Viscosity solutions for degenerate and nonmonotone elliptic equations, edited by B. da Vega, A. Sequeira and J. Videman. Plenum Press, New York & London, Appl. Nonlinear Anal. (1999) 185-210.  Zbl0960.35040
  31. O.A. Ladyzhenskaya and N.N. Ural'tseva, Linear and quasilinear equations of elliptic type,Second edition, revised. Izdat. “Nauka” Moscow (1973). English translation by Academic Press.  
  32. G.M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations. Ann. Scuola Normale Superiore Pisa Ser. IV21 (1994) 497-522.  Zbl0839.35018
  33. P. Lindqvist, A nonlinear eigenvalue problem. Rocky Mountain J.23 (1993) 281-288.  Zbl0785.34050
  34. P. Lindqvist, On the equation div ( | u | p - 2 u ) + Λ | u | p - 2 u =0. Proc. Amer. Math. Soc.109 (1990) 157-164 .  Zbl0714.35029
  35. P. Lindqvist, Addendum to "On the equation div ( | u | p - 2 u ) + Λ | u | p - 2 u =0". Proc. Amer. Math. Soc.116 (1992) 583-584.  Zbl0787.35027
  36. P. Lindqvist, Some remarkable sine and cosine functions. Ricerche Mat.44(1995) 269-290.  Zbl0944.33002
  37. J.L. Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969).  
  38. M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation. Comm. Partial Differential Equations22 (1997) 381-411.  Zbl0990.35077
  39. M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal.76 (1988) 140-159.  Zbl0662.35047
  40. S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Normale Superiore Pisa14 (1987) 404-421.  Zbl0665.35025
  41. G. Talenti, Personal Communication, letter dated Oct. 15, 2001  
  42. P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations51 (1984) 126-150.  Zbl0488.35017
  43. N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math.20 (1967) 721-747.  Zbl0153.42703
  44. N.N. Ural'tseva and A.B. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad Univ. Math.16 (1984) 263-270.  Zbl0569.35029
  45. I.M. Višik, Sur la résolutions des problèmes aux limites pour des équations paraboliques quasi-linèaires d'ordre quelconque. Mat. Sbornik59 (1962) 289-325.  
  46. I.M. Višik, Quasilinear strongly elliptic systems of differential equations in divergence form. Trans. Moscow. Math. Soc. 12 (1963) 140-208; Translation from Tr. Mosk. Mat. Obs. 12 (1963) 125-184.  

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