Existence of a unique solution to an elliptic partial differential equation when the average value is known

The purpose of this paper is to prove the existence of a unique classical solution $u(\bf{x})$ to the quasilinear elliptic partial differential equation $\nabla \cdot(a(u) \nabla u) = f$ for $\bf{x} \in \Omega$, which satisfies the condition that the average value $\frac{1}{|\Omega|}\int_{\Omega} u d\bf{x} = u_0$, where $u_0$ is a given constant and $\frac{1}{|\Omega|}\int_{\Omega} f d\bf{x} = 0$. Periodic boundary conditions will be used. That is, we choose for our spatial domain the N-dimensional torus $\mathbb{T}^N$, where $N = 2$ or $N = 3$. The key to the proof lies in obtaining a priori estimates for $u$.