Problem Let $D\subset \mathbb{R}^3$ be a smooth, connected, bounded domain, consider Dirichlet boundary problem \begin{equation} \left\{ \begin{aligned} &\Delta u=0, &\mathrm{in}\ \ \mathbb{R}^3-\overline{D} \\ &u=1, &\mathrm{on}\ \ \partial D \\ &u\rightarrow 0, & \mathrm{as} \ \ |x|\rightarrow \infty \end{aligned}\right. \end{equation} Prove that $u\in L^6(\mathbb{R}^3-\overline{D})$.
Attempt Using layer potential technique, there exists $\phi\in L^2(\partial D)$ such that $u=S\phi$, where $S$ is single layer potential, i.e. $$u(x)=-\int_{\partial D} \frac{\phi(y)}{4\pi |x-y|}\,dy \ \ \ \ x\in \mathbb{R}^3-\overline{D}$$ then by Green's identity $$\int_{\mathbb{R}^3-\overline{D}} |\nabla u|^2 =\int_{\partial D} \partial_Nu\cdot u=\int_{\partial D} \partial_Nu=\int_{\partial D}\phi<\infty$$ the last equality use jump relation for single layer potential, so that $\nabla u\in L^2(\mathbb{R}^3-\overline{D})$. But I do not know how to estimate $u\in L^6(\mathbb{R}^3-\overline{D})$, any help will be appreciated.