The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to \infty$
ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 1, p. 28-52

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 $p\to \infty$.

DOI : https://doi.org/10.1051/cocv:2003035
Classification:  35P30,  35B30,  49R50,  35P15
Keywords: eigenvalue, anisotropic, pseudo-Laplace, viscosity solution, minimal Lipschitz extension, concavity, symmetry, convex rearrangement
@article{COCV_2004__10_1_28_0,
author = {Belloni, Marino and Kawohl, Bernd},
title = {The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as ${p\rightarrow \infty }$},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {10},
number = {1},
year = {2004},
pages = {28-52},
doi = {10.1051/cocv:2003035},
zbl = {1092.35074},
mrnumber = {2084254},
language = {en},
url = {http://www.numdam.org/item/COCV_2004__10_1_28_0}
}

Belloni, Marino; Kawohl, Bernd. The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as ${p\rightarrow \infty }$. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 1, pp. 28-52. doi : 10.1051/cocv:2003035. http://www.numdam.org/item/COCV_2004__10_1_28_0/

[1] W. Allegretto and Yin Xi Huang, A Picone's identity for the $p$-Laplacian and applications. Nonlin. Anal. TMA 32 (1998) 819-830. | Zbl 0930.35053

[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. | Numdam | MR 1441395 | Zbl 0877.35040

[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. | MR 920052 | Zbl 0633.35061

[4] A. Anane, A. Benazzi and O. Chakrone, Sur le spectre d'un opérateur quasilininéaire elliptique “dégénéré”. Proyecciones 19 (2000) 227-248.

[5] G. Aronsson, Extension of functions satisfying Lipschitz conditions. Ark. Math. 6 (1967) 551-561. | MR 217665 | Zbl 0158.05001

[6] G. Aronsson, On the partial differential equation ${u}_{x}^{2}{u}_{xx}+2{u}_{x}{u}_{y}{u}_{xy}+{u}_{y}^{2}{u}_{yy}=0$. Ark. Math. 7 (1968) 395-425. | MR 237962 | Zbl 0162.42201

[7] G. Barles, Remarks on uniqueness results of the first eigenvalue of the $p$-Laplacian. Ann. Fac. Sci. Toulouse 9 (1988) 65-75. | Numdam | MR 971814 | Zbl 0621.35068

[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. | MR 1876420 | Zbl 0997.35023

[9] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the $p$-Laplace operator. Manuscripta Math. 109 (2002) 229-231. | MR 1935031 | Zbl 1100.35032

[10] T. Bhattacharya, E. Dibenedetto and J. Manfredi, Limits as $p\to \infty$ of ${\Delta }_{p}{u}_{p}=f$ 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 Equations 2001 (2001) 1-8. | MR 1836812 | Zbl 0966.35052

[12] H. Brezis and L. Oswald, Remarks on sublinear problems. Nonlinear Anal. 10 (1986) 55-64. | MR 820658 | Zbl 0593.35045

[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. | MR 1861094 | Zbl 0996.49019

[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. | Zbl 0755.35015

[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. | MR 1100211 | Zbl 0696.35087

[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. | MR 916325 | Zbl 0656.35039

[17] E. Dibenedetto, ${C}^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. TMA 7 (1983) 827-850. | MR 709038 | Zbl 0539.35027

[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. | MR 680591 | Zbl 0511.34006

[19] N. Fukagai, M. Ito and K. Narukawa, Limit as $p\to \infty$ of $p$-Laplace eigenvalue problems and ${L}^{\infty }$ inequality of the Poincaré type. Differ. Integral Equations 12 (1999) 183-206. | MR 1672746 | Zbl 1064.35512

[20] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31-46. | MR 666107 | Zbl 0494.49031

[21] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of second Order. Springer Verlag, Berlin-Heidelberg-New York (1977). | MR 473443 | Zbl 0361.35003

[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. | MR 1871409 | Zbl 1030.35088

[23] U. Janfalk, Behaviour in the limit, as $p\to \infty$, of minimizers of functionals involving $p$-Dirichlet integrals. SIAM J. Math. Anal. 27 (1996) 341-360. | MR 1377478 | Zbl 0853.35028

[24] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993) 51-74. | MR 1218686 | Zbl 0789.35008

[25] P. Juutinen, Personal Communications.

[26] P. Juutinen, P. Lindqvist and J. Manfredi, The $\infty$-eigenvalue problem. Arch. Rational Mech. Anal. 148 (1999) 89-105. | MR 1716563 | Zbl 0947.35104

[27] B. Kawohl, Rearrangements and convexity of level sets in PDE. Springer, Lecture Notes in Math. 1150 (1985). | MR 810619 | Zbl 0593.35002

[28] B. Kawohl, A family of torsional creep problems. J. Reine Angew. Math. 410 (1990) 1-22. | MR 1068797 | Zbl 0701.35015

[29] B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete Contin. Dynam. Systems 6 (2000) 683-690. | MR 1757396 | Zbl pre01492653

[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. | MR 1727452 | Zbl 0960.35040

[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. IV 21 (1994) 497-522. | Numdam | MR 1318770 | Zbl 0839.35018

[33] P. Lindqvist, A nonlinear eigenvalue problem. Rocky Mountain J. 23 (1993) 281-288. | MR 1212743 | Zbl 0785.34050

[34] P. Lindqvist, On the equation div${\left(|\nabla u|}^{p-2}{\nabla u\right)+\Lambda |u|}^{p-2}u$ $=0$. Proc. Amer. Math. Soc. 109 (1990) 157-164 . | MR 1007505 | Zbl 0714.35029

[35] P. Lindqvist, Addendum to “On the equation div${\left(|\nabla u|}^{p-2}{\nabla u\right)+\Lambda |u|}^{p-2}u$ $=0$. Proc. Amer. Math. Soc. 116 (1992) 583-584. | Zbl 0787.35027

[36] P. Lindqvist, Some remarkable sine and cosine functions. Ricerche Mat. 44 (1995) 269-290. | MR 1469702 | Zbl 0944.33002

[37] J.L. Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[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. | MR 1443043 | Zbl 0990.35077

[39] M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76 (1988) 140-159. | MR 923049 | Zbl 0662.35047

[40] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Normale Superiore Pisa 14 (1987) 404-421. | Numdam | MR 951227 | Zbl 0665.35025

[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 Equations 51 (1984) 126-150. | MR 727034 | Zbl 0488.35017

[43] N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721-747. | MR 226198 | Zbl 0153.42703

[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. | Zbl 0569.35029

[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] 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. | MR 156085 | Zbl 0144.36201