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

top
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

top

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

top
  1. W. Allegretto and Yin Xi Huang, A Picone's identity for the p-Laplacian and applications. Nonlin. Anal. TMA32 (1998) 819-830.  
  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.  
  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.  
  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.  
  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.  
  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.  
  9. M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator. Manuscripta Math.109 (2002) 229-231.  
  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.  
  12. H. Brezis and L.Oswald, Remarks on sublinear problems. Nonlinear Anal.10 (1986) 55-64.  
  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.  
  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.  
  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.  
  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.  
  17. E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA7 (1983) 827-850.  
  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.  
  20. M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math.148 (1982) 31-46.  
  21. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of second Order. Springer Verlag, Berlin-Heidelberg-New York (1977).  
  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.  
  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.  
  24. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal.123 (1993) 51-74.  
  25. P. Juutinen, Personal Communications.  
  26. P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem. Arch. Rational Mech. Anal.148 (1999) 89-105.  
  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.  
  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.  
  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.  
  33. P. Lindqvist, A nonlinear eigenvalue problem. Rocky Mountain J.23 (1993) 281-288.  
  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 .  
  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.  
  36. P. Lindqvist, Some remarkable sine and cosine functions. Ricerche Mat.44(1995) 269-290.  
  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.  
  39. M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal.76 (1988) 140-159.  
  40. S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Normale Superiore Pisa14 (1987) 404-421.  
  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.  
  43. N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math.20 (1967) 721-747.  
  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.  
  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.  

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.