On the Variational Inequalities for Certain Convex Function Classes

The existence and the C^1,α regularity of the weak solution to the variation inequality -(a_i(x, u, ∇u))_xi - (g_i(x, u))_xi + b(x, u, ∇u) ≥ 0 with respect to a closed convex function class is proved. For the regularity, we use the fact that the regularity for the viscosity solutions to the Hamilton-Jacobi equations implies the C^{1, α} interior regularity of the solution to the bilateral obstacle problem which in turn gives that of the solution to the variational inequality.