Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation
viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation
Summary:
In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.
References:
[1] Barles, G.: A weak Bernstein method for fully nonlinear elliptic equations. Differential Integral Equations 4 (2) (1991), 241–262. MR 1081182
[2] Chen, Y. G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991), 749–786. MR 1100211 | Zbl 0696.35087
[3] Crandall, M., Kocan, M., Lions, P. L., Swiȩch, A.: Existence results for uniformly elliptic and parabolic fully nonlinear equations. Electron. J. Differential Equations 24 (1999), 1–20.
[4] Crandall, M. G., Ishii, H., Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992), 1–67. DOI 10.1090/S0273-0979-1992-00266-5 | MR 1118699 | Zbl 0755.35015
[5] Crandall, M. G., Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42. DOI 10.1090/S0002-9947-1983-0690039-8 | MR 0690039 | Zbl 0599.35024
[6] Ishii, H., Lions, P. L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990), 26–78. DOI 10.1016/0022-0396(90)90068-Z | MR 1031377 | Zbl 0708.35031
[7] Kawohl, B., Kutev, N.: Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. Funkcial. Ekvac. 43 (2000), 241–253. MR 1795972 | Zbl 1142.35315
[8] Tersenov, Al., Tersenov, Ar.: Viscosity solutions of $p$-Laplace equation with nonlinear source. Arch. Math. (Basel) 88 (3) (2007), 259–268. DOI 10.1007/s00013-006-1873-9 | MR 2305604
[9] Trudinger, N. S.: Holder gradient estimates for fully nonlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 57–65. MR 0931007
Search
Partner of
