Abstract
In this paper we construct convex solutions for certain elliptic boundary value problems via Perron’s method. The solutions constructed are weak solutions in the viscosity sense, and our construction follows work of Ishii (Duke Math. J., 55 (2) 369–384, 1987). The same general approach appears in work of Andrews and Feldman (J. Differential Equations, 182 (2) 298–343, 2002) in which they show existence for a weak nonlocal parabolic flow of convex curves. The time independent special case of their work leads to a one dimensional elliptic result which we extend to two dimensions. Similar results are required to extend their theory of nonlocal geometric flows to surfaces.
The two dimensional case is essentially different from the one dimensional case and involves a regularity result (cf. Theorem 3.1), which has independent interest. Roughly speaking, given an arbitrary convex function (which is not smooth) supported at one point by a smooth function of prescribed Hessian (which is not convex), one must construct a third function that is both convex and smooth and appropriately approximates both of the given functions.
Keywords: Viscosity solutions, Elliptic partial differential equations, Perron procedure, Convexity, Regularity, Fully nonlinear, Monge-Ampere
Mathematics Subject Classification (2000:) 35J60, 53A05, 52A15, 26B05
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
1. Alvarez, O., Lasry, J.M., Lions, P.L.: Convex viscosity solutionsand state constraints. J. Math. Pures Appl. 9, no. 3, 265–288 (1997)
2. Andrews, B., Feldman, M.: Nonlocal geometric expansion of convex planar curves. J. Differential Equations 182, no. 2, 298–343 (2002)
3. Crandall, M., Ishii, H., Lions, P.L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, no. 1, 1–67 (1992)
4. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Second edition. Springer–Verlag, Berlin (1983)
5. Ishii, H.: Perron's method for Hamilton–Jacobi equations. Duke Math. J. 55, no. 2, 369–384 (1987)
6. Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rat. Mech. Anal. 101, 1–27 (1988)
7. Jensen, R.: Uniquess criteria for viscosity solutions of fully nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 38, 629–667 (1989)
8. Schneider, R.: Convex Bodies: The Brunn–Monkowski Theory. Encyclopedia of mathematics and its applications, vol. 44. Cambridge University Press, Cambridge (1993)
9. Soner, H.M.: Motion of a set by the curvature of its boundary. J. Differential Equations 101, no. 2, 313–372 (1993)
Author information
Authors and Affiliations
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Feldman, M., McCuan, J. Constructing convex solutions via Perron’s method. Ann. Univ. Ferrara 53, 65–94 (2007). https://doi.org/10.1007/s11565-007-0006-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11565-007-0006-0