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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; belloni@math.unipr.it.
Bernd Kawohl
Affiliation:
Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany; kawohl@mi.uni-koeln.de..
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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

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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.