W2,ploc(Ω)\cap C1,α(\bar Ω) Viscosity Solutions of Neumann Problems for Fully Nonlinear Elliptic Equations (Journal of Partial Differential Equations, 1995)
In this paper we study fully nonlinear elliptic equations F(D²u, x) = 0 in Ω ⊂ R^n with Neumann boundary conditions \frac{∂u}{∂v} = a(x)u under the rather mild structure conditions and without the concavity condition. We establish the global C^{1,Ω} estimates and the interior W^{2,p} estimates for W^{2,q}(Ω) solutions (q > 2n) by introducing new independent variables, and moreover prove the existence of W^{2,p}_{loc}(Ω)∩ C^{1,α}(\bar Ω} viscosity solutions by using the accretive operator methods, where p E (0, 2), α ∈ (0, 1}.
Keywords: Viscosity solutions; Neumann boundary conditions; fully nonlinear equations; global C^{1; α} estimates; interior W^{2; p} estimates