On the Variational Inequalities for Certain Convex Function Classes

The existence and the C1, α regularity of the weak solution to the variation inequality −(ai(x, u, u))xi − (gi(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 C1, α interior regularity of the solution to the bilateral obstacle problem which in turn gives that of the solution to the variational inequality.