On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains. (English) Zbl 0736.35044

Dupuis, Paul; Ishii, Hitoshi

Nonlinear Anal., Theory Methods Appl. 15, No. 12, 1123-1138 (1990).

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.

Reviewer: J.Heinonen (Ann Arbor)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

oblique derivative; existence of viscosity sub- and super-solutions

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