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

Marino Belloni; Bernd Kawohl

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • 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 ${p\rightarrow \infty }$." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2004): 28-52. <http://eudml.org/doc/245898>.

@article{Belloni2004,
abstract = {We consider the pseudo-$p$-laplacian, an anisotropic version of the $p$-laplacian operator for $p\ne 2$. We study relevant properties of its first eigenfunction for finite $p$ and the limit problem as $p\rightarrow \infty $.},
author = {Belloni, Marino, Kawohl, Bernd},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {eigenvalue; anisotropic; pseudo-Laplace; viscosity solution; minimal Lipschitz extension; concavity; symmetry; convex rearrangement; pseudo--Laplacian operator},
language = {eng},
number = {1},
pages = {28-52},
publisher = {EDP-Sciences},
title = {The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $\{p\rightarrow \infty \}$},
url = {http://eudml.org/doc/245898},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Belloni, Marino
AU - Kawohl, Bernd
TI - The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as ${p\rightarrow \infty }$
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
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 $p\ne 2$. We study relevant properties of its first eigenfunction for finite $p$ and the limit problem as $p\rightarrow \infty $.
LA - eng
KW - eigenvalue; anisotropic; pseudo-Laplace; viscosity solution; minimal Lipschitz extension; concavity; symmetry; convex rearrangement; pseudo--Laplacian operator
UR - http://eudml.org/doc/245898
ER -

References

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  1. [1] W. Allegrettoand Yin Xi Huang, A Picone’s identity for the p -Laplacian and applications. Nonlin. Anal. TMA 32 (1998) 819-830. Zbl0930.35053
  2. [2] A. Alvino, V. Ferone, G. Trombetti and P.L. Lions, Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 275-293. Zbl0877.35040MR1441395
  3. [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. Zbl0633.35061MR920052
  4. [4] A. Anane, A. Benazzi and O. Chakrone, Sur le spectre d’un opérateur quasilininéaire elliptique “dégénéré”. Proyecciones 19 (2000) 227-248. 
  5. [5] G. Aronsson, Extension of functions satisfying Lipschitz conditions. Ark. Math. 6 (1967) 551-561. Zbl0158.05001MR217665
  6. [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. Zbl0162.42201MR237962
  7. [7] G. Barles, Remarks on uniqueness results of the first eigenvalue of the p -Laplacian. Ann. Fac. Sci. Toulouse 9 (1988) 65-75. Zbl0621.35068MR971814
  8. [8] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Partial Differential Equations 26 (2001) 2323-2337. Zbl0997.35023MR1876420
  9. [9] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p -Laplace operator. Manuscripta Math. 109 (2002) 229-231. Zbl1100.35032MR1935031
  10. [10] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p of Δ p u p = f and related extremal problems. Rend. Sem. Mat., Fasciolo Speciale Nonlinear PDE’s. Univ. Torino (1989) 15-68. 
  11. [11] T. Bhattacharya, An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions. Electron. J. Differential Equations 2001 (2001) 1-8. Zbl0966.35052MR1836812
  12. [12] H. Brezis and L. Oswald, Remarks on sublinear problems. Nonlinear Anal. 10 (1986) 55-64. Zbl0593.35045MR820658
  13. [13] M.G. Crandall, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differential Equations 13 (2001) 123-139. Zbl0996.49019MR1861094
  14. [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. [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. Zbl0696.35087MR1100211
  16. [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. Zbl0656.35039MR916325
  17. [17] E. DiBenedetto, C 1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA 7 (1983) 827-850. Zbl0539.35027MR709038
  18. [18] A. Elbert, A half-linear second order differential equation. Qualitative theory of differential equations, (Szeged 1979). Colloq. Math. Soc. János Bolyai 30 (1981) 153-180. Zbl0511.34006MR680591
  19. [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. Zbl1064.35512MR1672746
  20. [20] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31-46. Zbl0494.49031MR666107
  21. [21] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of second Order. Springer Verlag, Berlin-Heidelberg-New York (1977). Zbl0361.35003MR473443
  22. [22] T. Ishibashi and S. Koike, On fully nonlinear pdes derived from variational problems of L p -norms. SIAM J. Math. Anal. 33 (2001) 545-569. Zbl1030.35088MR1871409
  23. [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. Zbl0853.35028MR1377478
  24. [24] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993) 51-74. Zbl0789.35008MR1218686
  25. [25] P. Juutinen, Personal Communications. 
  26. [26] P. Juutinen, P. Lindqvist and J. Manfredi, The -eigenvalue problem. Arch. Rational Mech. Anal. 148 (1999) 89-105. Zbl0947.35104MR1716563
  27. [27] B. Kawohl, Rearrangements and convexity of level sets in PDE. Springer, Lecture Notes in Math. 1150 (1985). Zbl0593.35002MR810619
  28. [28] B. Kawohl, A family of torsional creep problems. J. Reine Angew. Math. 410 (1990) 1-22. Zbl0701.35015MR1068797
  29. [29] B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete Contin. Dynam. Systems 6 (2000) 683-690. Zbl1157.35342MR1757396
  30. [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. Zbl0960.35040MR1727452
  31. [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. [32] G.M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations. Ann. Scuola Normale Superiore Pisa Ser. IV 21 (1994) 497-522. Zbl0839.35018MR1318770
  33. [33] P. Lindqvist, A nonlinear eigenvalue problem. Rocky Mountain J. 23 (1993) 281-288. Zbl0785.34050MR1212743
  34. [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 . Zbl0714.35029MR1007505
  35. [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. [36] P. Lindqvist, Some remarkable sine and cosine functions. Ricerche Mat. 44 (1995) 269-290. Zbl0944.33002MR1469702
  37. [37] J.L. Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969). Zbl0189.40603MR259693
  38. [38] M. Ohnuma and K. Sato, Singular degenerate parabolic equations with applications to the p -Laplace diffusion equation. Comm. Partial Differential Equations 22 (1997) 381-411. Zbl0990.35077MR1443043
  39. [39] M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76 (1988) 140-159. Zbl0662.35047MR923049
  40. [40] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Normale Superiore Pisa 14 (1987) 404-421. Zbl0665.35025MR951227
  41. [41] G. Talenti, Personal Communication, letter dated Oct. 15, 2001 
  42. [42] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984) 126-150. Zbl0488.35017MR727034
  43. [43] N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721-747. Zbl0153.42703MR226198
  44. [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. [45] I.M. Višik, Sur la résolutions des problèmes aux limites pour des équations paraboliques quasi-linèaires d’ordre quelconque. Mat. Sbornik 59 (1962) 289-325. 
  46. [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. Zbl0144.36201MR156085

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