# Uniqueness of the exterior Neumann problem

I have a question regarding the uniqueness of the exterior Neumann problem, as stated by Lemma 2.4 in this paper.

Let $$\Omega \subset \mathbb{R}^3$$ be a bounded domain with $$C^2$$ boundary $$\partial\Omega$$ and let $$\Omega^\prime = \mathbb{R}^3 \setminus \overline{\Omega}$$. The exterior Neumann problem

\begin{align} \Delta \varphi(x) &= 0 \qquad &&\textrm{for} \; x \in \Omega^\prime \; , \\ \tag{1}\label{eq:NeumannProblem} \nu(x) \cdot \nabla\varphi(x) &= g(x) \qquad &&\textrm{for} \; x \in \partial\Omega \; , \\ \varphi(x) &\to 0 \qquad &&\textrm{for} \; \lvert x \rvert \to \infty \; , \end{align}

where $$\nu$$ is the outer normal w.r.t. $$\Omega$$ and $$g \in C^0(\partial\Omega, \mathbb{R})$$, has a unique solution $$\varphi \in C^2(\Omega, \mathbb{R}) \cap C^0(\bar{\Omega}, \mathbb{R})$$.

Uniqueness of the solution can be shown by realising that $$\varphi(x) \to 0$$ for $$\lvert x \rvert \to \infty$$ implies the stronger decay

\begin{aligned} \lvert \varphi(x) \rvert = \mathcal{O}(\lvert x \rvert^{-1}) \quad \textrm{for} \; \lvert x \rvert \; \to \infty \; , \\ \lvert \nabla\varphi(x) \rvert = \mathcal{O}(\lvert x \rvert^{-2}) \quad \textrm{for} \; \lvert x \rvert \; \to \infty \; , \end{aligned}

and by using Green's first identity (see, e.g. The Gauss-Green theorem for unbounded domain).

Consider the slightly modified exterior Neumann problem

\begin{align} \Delta \phi(x) &= 0 \qquad &&\textrm{for} \; x \in \Omega^\prime \; , \\ \tag{2}\label{eq:NeumannProblem2} \nu(x) \cdot \nabla\phi(x) &= 0 \qquad &&\textrm{for} \; x \in \partial\Omega \; , \\ \nabla\phi(x) &\stackrel{\mathrm{unif.}}{\to} u_\infty \qquad &&\textrm{for} \; \lvert x \rvert \to \infty \; , \end{align}

where $$u_\infty \in \mathbb{R}^3$$. The solution of the exterior Neumann problem \eqref{eq:NeumannProblem2} is given by

$$$$\phi(x) = \varphi(x) + u_\infty \cdot x$$$$

where $$\varphi$$ is the solution to the exterior Neumann problem \eqref{eq:NeumannProblem} with $$g(x) = - \nu(x) \cdot u_\infty$$.

According to Lemma 2.4 of this paper, the exterior Neumann problem \eqref{eq:NeumannProblem2} has a unique solution. But with the standard proof, using Green's first identity applied to the difference $$\psi$$ of two solutions of the exterior Neumann problem \eqref{eq:NeumannProblem2}, one would need to show that

$$$$\tag{3}\label{eq:Limit} \int_{B_R(0)} \psi \left( n \cdot \nabla \psi \right) \mathrm{d}S \to 0 \qquad \textrm{for} \; R \to \infty \; ,$$$$

where $$n$$ is the outer normal w.r.t. the ball $$B_R(0)$$ of radius $$R$$ (which is sufficiently large to contain $$\Omega$$).

To my understanding we can only infer that $$\lvert \nabla\psi(x) \rvert = \mathcal{O}(\lvert x \rvert^{-1})$$ (due to $$\nabla\psi(x) \stackrel{\mathrm{unif.}}{\to} 0$$), which is not sufficient to prove \eqref{eq:Limit}. Is there a stronger decay of $$\psi$$ related to the uniform convergence towards $$0$$ that I fail to see or what am I missing?

Any other reference elaborating on the exterior Neumann problem \eqref{eq:NeumannProblem2}, that is for the specific limiting behaviour $$\nabla\phi \to u_\infty$$, is appreciated.