Abstract
We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.
Article PDF
Similar content being viewed by others
References
Alvarez O., Lasry J.-M., Lions P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997)
Brascamp H.J., Lieb E.H.: On extensions of the Brunn–Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
Caffarelli L.A., Spruck J.: Convexity properties of solutions to some classical variational problems. Commun. Partial Differ. Equ. 7(11), 1337–1379 (1982)
Crandall M.G., Evans L.C., Gariepy R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)
Crandall M.G., Ishii H.: The maximum principle for semicontinuous functions. Differ. Integral Equ. 3(6), 1001–1014 (1990)
Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Giga Y.: Surface evolution equations. A level set approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel (2006)
Juutinen P.: Principal eigenvalue of a very badly degenerate operator and applications. J. Differ. Equ. 236(2), 532–550 (2007)
Juutinen P., Lindqvist P., Manfredi J.J.: The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)
Kawohl B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985)
Kawohl B.: A remark on N. Korevaar’s concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem. Math. Methods Appl. Sci. 8(1), 93–101 (1986)
Kennington A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34(3), 687–704 (1985)
Korevaar N.J.: Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32(1), 73–81 (1983)
Korevaar N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32(4), 603–614 (1983)
Lindqvist P., Manfredi J., Saksman E.: Superharmonicity of nonlinear ground states. Rev. Mat. Iberoamericana 16(1), 17–28 (2000)
Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 3, 403–421 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is supported by the Academy of Finland, project 129784.
Rights and permissions
About this article
Cite this article
Juutinen, P. Concavity maximum principle for viscosity solutions of singular equations. Nonlinear Differ. Equ. Appl. 17, 601–618 (2010). https://doi.org/10.1007/s00030-010-0071-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-010-0071-4