zbMATH — the first resource for mathematics

On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains. (English) Zbl 0736.35044
The fully nonlinear elliptic \(PDE\) \(F(x,u,Du,D^ 2u)=0\) in \(\Omega\) with the oblique derivative condition \(\partial u/\partial\gamma+f(x,u)=0\) on \(\partial\Omega\) is known to have unique solutions in domains \(\Omega\) that are sufficiently smooth, see P. L. Lions [Duke Math. J. 52, 793-820 (1985; Zbl 0599.35025)]; a basic assumption on \(F\) here is the degenerate ellipticity. This paper provides the existence of viscosity sub- and super-solutions \(u\) and \(v\) together with the inequality \(u\leq v\) in domains which are not necessarily smooth. This and Perron’s method guarantee the existence of a unique solution.

35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI
[1] Crandall, M.G., Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order, Ann. inst. H. Poincaré analyse non linéaire, 6, 419-435, (1989) · Zbl 0734.35033
[2] Crandall, M.G.; Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. am. math. soc., 277, 1-42, (1983) · Zbl 0599.35024
[3] Dupuis, P.; Ishii, H.; Soner, H.M., A viscosity solution approach to the asymptotic analysis of queueing systems, Ann. prob., 18, 226-255, (1990) · Zbl 0715.60035
[4] \scFleming W. H., \scIshii H. & \scMenaldi J.-L. (in preparation).
[5] Ishii, H., Perron’s method for Hamilton-Jacobi equations, Duke math. J., 55, 369-384, (1987) · Zbl 0697.35030
[6] Ishii, H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. scuola norm. sup. Pisa, 16, 105-135, (1989) · Zbl 0701.35052
[7] Ishii, H., On uniqueness and existence of solutions of fully nonlinear elliptic pdes, Communs pure appl. math., 42, 15-45, (1989) · Zbl 0645.35025
[8] Ishii, H.; Lions, P.-L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. diff. eqns, 83, 26-78, (1990) · Zbl 0708.35031
[9] Jensen, R., The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Archs ration. mech. analysis, 101, 1-27, (1988) · Zbl 0708.35019
[10] Jensen, R., Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations, Indiana univ. math. J., 38, 629-667, (1989) · Zbl 0838.35037
[11] Jensen, R.; Lions, P.-L.; Souganidis, P.E., A uniqueness result for viscosity solutions of second-order fully nonlinear partial differential equations, Proc. am. math. soc., 102, 975-978, (1988) · Zbl 0662.35048
[12] Lions, P.-L., Optimal control of diffusion processes, part 2: viscosity solutions and uniqueness, Communs partial diff. eqns, 8, 1229-1276, (1983) · Zbl 0716.49023
[13] Lions, P.-L., Neumann type boundary conditions for Hamilton-Jacobi equations, Duke math. J., 52, 793-820, (1985) · Zbl 0599.35025
[14] Lions, P.-L.; Trudinger, N.S., Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation, Math. Z., 191, 1-15, (1986) · Zbl 0593.35046
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.