Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. (English) Zbl 0708.35031

Ishii, H.; Lions, P. L.

J. Differ. Equations 83, No. 1, 26-78 (1990).

This is a long and technical paper about viscosity solutions for fully nonlinear elliptic equations \(F(x,u,\nabla u,D^ 2u)=0\) in \(\Omega\) under various boundary conditions. The common strategy to tackle these equations is the observation of the first author [Duke Math. J. 55, 362- 384 (1987) and Commun. Pure Appl. Math. 42, No., 15-45 (1989; Zbl 0645.35025)] that (unique) existence is implied by a Perron-process, if viscosity sub- and supersolutions are known and a kind of maximum principle can be proved. It reads as follows: Whenever u (resp. v) is an usc (resp. lsc) bounded viscosity sub- (resp. super-) solution, then \[ u-v\leq \sup_{x\in \partial \Omega}\{u^*(x)-v_*(x)\}^+\text{ in } \Omega \] \(\sup_{y\in \Omega,y\to x} \sup u(y)\) and \(v_*=-(-v)^*.\)
Hence the problem remains in (and most of the paper is devoted to) veryfying this under various structure conditions on F, including Isaac- Bellman equations and also Monge-Ampère equations. See also R. Jensen [Arch. Ration. Mech. Anal. 101, No.1, 1-27 (1988)]. The paper closes with some remarks to the regularity of solutions. For \(C_{1,\alpha}\)-estimates, see also N. S. Trudinger [Proc. R. Soc. Edinb. Sect. A 108, No.1/2, 57-65 (1988; Zbl 0653.35026)].

Reviewer: M.Wiegner

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B50	Maximum principles in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs
35J70	Degenerate elliptic equations
35D10	Regularity of generalized solutions of PDE (MSC2000)

Keywords:

viscosity solutions; fully nonlinear elliptic equations; Perron-process; Isaac-Bellman equations; Monge-Ampère equations

Citations:

Zbl 0645.35025; Zbl 0653.35026

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