A remark on the existence of viscosity solutions for quasilinear elliptic equations. (English) Zbl 1103.35033
The paper deals with viscosity solutions to \(-\sum_{i,j=1}^n a_{ij}(x)u_{x_i x_j}+b(x,u,\nabla u)=0\) in a bounded, open domain \(\emptyset\neq\Omega\subset{\mathbb R}^n\) subject to a homogeneous Dirichlet boundary condition. The functions \(a, b\) are continuous, and the problem is supposed to be uniformly elliptic and proper in the sense of M. G. Crandall, H. Ishii and P.-L. Lions [Bull. Am. Math. Soc., New Ser. 27, No. 1, 1–67 (1992; Zbl 0755.35015)]. The author provides hypotheses under which Perron’s method for viscosity solutions applies, and obtains existence and uniqueness results in this context.
Reviewer: Georg Hetzer (Auburn)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
Citations:
Zbl 0755.35015References:
| [1] | Barles, G., A weak Bernstein method for fully nonlinear elliptic equations, Differential and Integr. Equations, 4, 2, 241-262 (1991) · Zbl 0733.35014 |
| [2] | Caffarelli, L. A., Interior a priori estimates for solutions of fully nonlinear elliptic equations, Ann. Math., 130, 189-213 (1989) · Zbl 0692.35017 |
| [3] | Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015 |
| [4] | Crandall, M. G.; Kocan, M.; Lions, P. L.; Swiech, A., Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24, 1-20 (1999) · Zbl 0927.35029 |
| [5] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin, Heidelberg, New York · Zbl 0691.35001 |
| [6] | Ishii, H.; Lions, P. L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83, 26-78 (1990) · Zbl 0708.35031 |
| [7] | Li, J.; Liu, J.; Yu, Q., Uniqueness and existence of viscosity solutions of the Dirichlet boundary problem for nonlinear elliptic second order equations, J. Math. Study, 35, 1, 26-31 (2002) · Zbl 1142.35426 |
| [8] | Miller, K., Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl., 76, 93-105 (1967) · Zbl 0149.32101 |
| [9] | Miller, K., Extremal barriers on cones with Phragmen-Lindelof theorems and other applications, Ann. Mat. Pura Appl., 90, 297-329 (1971) · Zbl 0231.35004 |
| [10] | Safonov, M. V., On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk SSSR Ser. Mat., 52, 1272-1287 (1988), (in Russian) (translation in Math. USSR-Izv. 33 (1989) 597-612) · Zbl 0682.35048 |
| [11] | Swiech, A., \(W^{1, p}\)-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2, 1005-1027 (1997) · Zbl 1023.35509 |
| [12] | Trudinger, N. S., Holder gradient estimates for fully nonlinear elliptic equations, Proc. R. Soc. Edinburgh, 108 A, 57-65 (1988) · Zbl 0653.35026 |
| [13] | Yi, Q.; Zhao, J. N., Regularity of viscosity solutions of the Dirichlet boundary problem for a class of quasilinear degenerate parabolic equations, Chinese J. Contemp. Math., 23, 1, 63-74 (2002) · Zbl 1043.35093 |
| [14] | Zhu, R. J., Existence, uniqueness and regularity of viscosity solutions of fully nonlinear parabolic equations under natural structure conditions, Beijing Shifan Daxue Xuebao, 30, 3, 290-300 (1994) · Zbl 0821.35083 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
