The Obstacle Problems for Second Order Fully Nonlinear Elliptic Equations with Neumann Boundary Conditions

Abstract

In this paper we prove the existence theorem of the strong solutions to the obstacle problems for second order fully nonlinear elliptic equations with the Neumann boundary conditions F(x, u, Du, D²u) ≥ 0, x ∈ Ω u ≤ g, x ∈ Ω (u - g)F(x, u, Du, D²u) = 0, x ∈ Ω D_vu = φ(x, u), x ∈ ∂Ω where F(x, z, p, r) satisfies the natural structure conditions and is concave with respect to r, p, and φ(x, z) is nondecreasing in z, and g(x) satisfies the consistency condition.