Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well

A new type weighted reverse Poincaré inequality is established for a difference of two continuous weak subsolutions of a linear second order uniformly elliptic partial differential equation in the ball.This result is the key to deriving the error estimate for the gradient of the analytically unknown value function of the optimal stochastic control problem from the uniform error of the value function itself in the related numerical approximation problems.