Geodesic
Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations
Archivum mathematicum, Tome 45 (2009) no. 1, pp. 19-35
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 35D05, 35D40, 35J60, 49L25
Keywords: viscosity subsolution; viscosity supersolution; mean curvature equation; pseudo $p$-Laplace equation 
@article{ARM_2009_45_1_a1,
     author = {Tersenov, Aris S.},
     title = {Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations},
     journal = {Archivum mathematicum},
     pages = {19--35},
     year = {2009},
     volume = {45},
     number = {1},
     mrnumber = {2591658},
     zbl = {1212.35134},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a1/}
}
TY  - JOUR
AU  - Tersenov, Aris S.
TI  - Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations
JO  - Archivum mathematicum
PY  - 2009
SP  - 19
EP  - 35
VL  - 45
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a1/
LA  - en
ID  - ARM_2009_45_1_a1
ER  - 
%0 Journal Article
%A Tersenov, Aris S.
%T Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations
%J Archivum mathematicum
%D 2009
%P 19-35
%V 45
%N 1
%U http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a1/
%G en
%F ARM_2009_45_1_a1
Tersenov, Aris S. Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations. Archivum mathematicum, Tome 45 (2009) no. 1, pp. 19-35. http://geodesic.mathdoc.fr/item/ARM_2009_45_1_a1/

[1] Barles, G.: A weak Bernstein method for fully nonlinear elliptic equations. Differential Integral Equations 4 (2) (1991), 241–262. | MR

[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 | Zbl

[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 | MR | Zbl

[5] Crandall, M. G., Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42. | DOI | MR | Zbl

[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 | MR | Zbl

[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 | Zbl

[8] Tersenov, Al., Tersenov, Ar.: Viscosity solutions of $p$-Laplace equation with nonlinear source. Arch. Math. (Basel) 88 (3) (2007), 259–268. | DOI | MR

[9] Trudinger, N. S.: Holder gradient estimates for fully nonlinear elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), 57–65. | MR