Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T11:18:53.177Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 February 2004

Marino Belloni  and
Bernd Kawohl
Marino Belloni
Affiliation:
Dip. di Matematica, Universita di Parma, Via d'Azeglio 85, 43100 Parma, Italy; [email protected].
Bernd Kawohl
Affiliation:
Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany; [email protected]..
Get access

Abstract

We consider the pseudo-p-Laplacian, an anisotropicversion of the p-Laplacian operator for $p\not=2$ . We studyrelevant properties of its first eigenfunction for finite p andthe limit problem as p → ∞.

Keywords

Eigenvalueanisotropicpseudo-Laplaceviscosity solutionminimal Lipschitz extensionconcavitysymmetryconvex rearrangement
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 28 - 52
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allegretto, W. and Yin Xi Huang, A Picone's identity for the p-Laplacian and applications. Nonlin. Anal. TMA 32 (1998) 819-830. CrossRef
Alvino, A., Ferone, V., Trombetti, G. and Lions, P.L., Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 275-293. CrossRef
Anane, A., 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.
Anane, A., Benazzi, A. and Chakrone, O., Sur le spectre d'un opérateur quasilininéaire elliptique "dégénéré". Proyecciones 19 (2000) 227-248.
Aronsson, G., Extension of functions satisfying Lipschitz conditions. Ark. Math. 6 (1967) 551-561. CrossRef
Aronsson, G., On the partial differential equation $u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0$ . Ark. Math. 7 (1968) 395-425. CrossRef
Barles, G., Remarks on uniqueness results of the first eigenvalue of the p-Laplacian. Ann. Fac. Sci. Toulouse 9 (1988) 65-75. CrossRef
Barles, G. and Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Partial Differential Equations 26 (2001) 2323-2337. CrossRef
Belloni, M. and Kawohl, B., A direct uniqueness proof for equations involving the p-Laplace operator. Manuscripta Math. 109 (2002) 229-231. CrossRef
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.
Bhattacharya, T., An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions. Electron. J. Differential Equations 2001 (2001) 1-8.
Brezis, H. and L.Oswald, Remarks on sublinear problems. Nonlinear Anal. 10 (1986) 55-64. CrossRef
Crandall, M.G., Evans, L.C. and Gariepy, R.F., Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differential Equations 13 (2001) 123-139.
Crandall, M.G., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. CrossRef
Chen, Y.G., Giga, Y. and Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33 (1991) 749-786. CrossRef
Diaz, J.I. and Saá, J.E., Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 521-524.
DiBenedetto, E., C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA 7 (1983) 827-850. CrossRef
Elbert, A., A half-linear second order differential equation. Qualitative theory of differential equations, (Szeged 1979). Colloq. Math. Soc. János Bolyai 30 (1981) 153-180.
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.
Giaquinta, M. and Giusti, E., On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31-46. CrossRef
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of second Order. Springer Verlag, Berlin-Heidelberg-New York (1977).
Ishibashi, T. and Koike, S., On fully nonlinear pdes derived from variational problems of Lp -norms. SIAM J. Math. Anal. 33 (2001) 545-569. CrossRef
Janfalk, U., Behaviour in the limit, as p → ∞, of minimizers of functionals involving p-Dirichlet integrals. SIAM J. Math. Anal. 27 (1996) 341-360. CrossRef
Jensen, R., Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993) 51-74. CrossRef
P. Juutinen, Personal Communications.
Juutinen, P., Lindqvist, P. and Manfredi, J., The ∞-eigenvalue problem. Arch. Rational Mech. Anal. 148 (1999) 89-105. CrossRef
B. Kawohl, Rearrangements and convexity of level sets in PDE. Springer, Lecture Notes in Math. 1150 (1985).
Kawohl, B., A family of torsional creep problems. J. Reine Angew. Math. 410 (1990) 1-22.
Kawohl, B., Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete Contin. Dynam. Systems 6 (2000) 683-690. CrossRef
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.
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.
Lieberman, G.M., Gradient estimates for a new class of degenerate elliptic and parabolic equations. Ann. Scuola Normale Superiore Pisa Ser. IV 21 (1994) 497-522.
Lindqvist, P., A nonlinear eigenvalue problem. Rocky Mountain J. 23 (1993) 281-288. CrossRef
Lindqvist, P., On the equation div $(|\nabla u|^{p-2}\nabla u)+ \Lambda |u|^{p-2}u$ =0. Proc. Amer. Math. Soc. 109 (1990) 157-164 .
Lindqvist, P., Addendum to "On the equation div $(|\nabla u|^{p-2}\nabla u)+ \Lambda |u|^{p-2}u$ =0". Proc. Amer. Math. Soc. 116 (1992) 583-584.
Lindqvist, P., Some remarkable sine and cosine functions. Ricerche Mat. 44(1995) 269-290.
J.L. Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969).
Ohnuma, M. and Sato, K., Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation. Comm. Partial Differential Equations 22 (1997) 381-411.
Ôtani, M., Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76 (1988) 140-159. CrossRef
Sakaguchi, S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Normale Superiore Pisa 14 (1987) 404-421.
G. Talenti, Personal Communication, letter dated Oct. 15, 2001
Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984) 126-150. CrossRef
Trudinger, N., Harnack, On type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721-747. CrossRef
Ural'tseva, N.N. and Urdaletova, A.B., The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad Univ. Math. 16 (1984) 263-270.
Višik, I.M., Sur la résolutions des problèmes aux limites pour des équations paraboliques quasi-linèaires d'ordre quelconque. Mat. Sbornik 59 (1962) 289-325.
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.