Existence of a unique solution to an elliptic partial differential equation - Diane L. Denny (Electronic Journal of Differential Equations, 2019)
The purpose of this article is to prove the existence of a unique classical solution to the quasilinear elliptic equation $-\nabla ·(a(u) \nabla u)=f$ for $x \in Ω$, which satisfies the condition that $u(x_0)=u_0$ at a given point $x_0 \in Ω$, under the boundary condition $n(x)· \nabla u(x)=0$ for $ x \in \partial Ω$ where $n(x)$ is the outward unit normal vector and where $\frac{1}{|Ω|}\int_{Ω} f\,dx=0$. The domain $Ω \subset \mathbb{R}^{N}$ is a bounded, connected, open set with a smooth boundary, and N=2 or N=3. The key to the proof lies in obtaining a priori estimates for the solution.