The regularity estimates for the solutions of general elliptic equations in Orlicz spaces

Let Ω be an open set in high dimension Euclidean spaces Rⁿ, and let L denote the second order elliptic operator in the general form, which will characterize many of the original phenomenon of nature. As a very useful space, Orlicz space L^Φ includes many classical spaces such as L^p space, where 1< p < ∞ and Φ is a convex increasing function. The purpose of this paper is to use the stopping time technique and establish the regularity estimates in Orlicz spaces for the second order derivatives of the solutions to the general second order elliptic equations and their Dirichlet problems. We will obtain that, if u is the solution of the Dirichlet problem to the equation Lu = f for any given data f, then the weak solution u as well as its gradient ∇u and second order derivatives ∇² u can be controlled by the data f even in the Orlicz spaces.