Existence of a unique solution to an elliptic partial differential equation

Abstract

The purpose of this article is to prove the existence of a unique classical solution to the quasilinear elliptic equation $-\nabla \cdot(a(u) \nabla u)=f$ for $\mathbf{x} \in \Omega$, which satisfies the condition that $u(\mathbf{x}_0)=u_0$ at a given point $\mathbf{x}_0 \in \Omega$, under the boundary condition $\mathbf{n}(\mathbf{x})\cdot \nabla u(\mathbf{x})=0$ for $ \mathbf{x} \in \partial \Omega$ where $\mathbf{n}(\mathbf{x})$ is the outward unit normal vector and where $\frac{1}{|\Omega|}\int_{\Omega} f\,d\mathbf{x}=0$. The domain $\Omega \subset \mathbb{R}^{N}$ 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.

Keywords

• Existence
• uniqueness
• quasilinear
• elliptic