Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 110, pp. 1-13.

Title: Existence of a unique solution to an elliptic partial
       differential equation

Author: Diane L. Denny (Texas A&M Univ., Corpus Christi, TX, USA)

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.

Submitted February 24, 2018. Published September 26, 2019.
Math Subject Classifications: 35A05
Key Words: Existence; uniqueness; quasilinear; elliptic.