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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{equation} \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} \end{equation}

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

\begin{equation} \phi(x) = \varphi(x) + u_\infty \cdot x \end{equation}

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

\begin{equation}\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 \; , \end{equation}

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.

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