A correction of Theorem 2.1 in [1] reads as follows.

### Theorem 2.1

*Let*
\(P_{1}\)*hold*. *Let*
\((\rho_{n})_{n=1}^{\infty}\in{\mathcal{A}}_{M}\)*be a minimizing sequence of F*. *Then there exists a subsequence*, *still denoted by*
\((\rho_{n} )_{n=1}^{\infty }\), *and a sequence of translations*
\(T\rho_{n}:=\rho_{n} (\cdot+a_{n} e_{3} ) \), *where*
\(a_{n} \)*are constants*, *and*
\(e_{3}=(0,0,1)\), *such that*

$$F(\rho_{0})=\inf_{{\mathcal{A}}_{M}}F (\rho) =h_{M}+ \rho_{0} $$

*and*
\(T\rho_{n} \rightharpoonup\rho_{0} \)*weakly in*
\(L^{\frac{4}{3}}({\mathbb{R}}^{3})\). *For the induced potentials*, *we have*
\(\nabla\Phi_{T\rho_{n} }\rightharpoonup \nabla\Phi_{\rho_{0} } \)*weakly in*
\(L^{2}({\mathbb{R}}^{3})\).

### Proof

Define

$$I_{lm}:= \int \int\frac{\rho_{l}(x) \rho_{m} (y)}{x-y}\,dy\,dx $$

for \(l,m=1,2,3\).

Let \(\rho=\rho_{1} + \rho_{2} + \rho_{3}\), where \(\rho_{1}=\chi _{B_{R_{1}}}\rho\), \(\rho_{2}=\chi_{B_{R_{1},R_{2}}}\rho\), and \(\rho_{3}=\chi_{B_{R_{2}}}\rho\). So we have

$$F (\rho)= F (\rho_{1})+ F (\rho_{2})+F ( \rho_{3})-I_{12}- I_{13}-I_{23}. $$

Choosing \(R_{2} >2R_{1}\), we have

$$I_{13}\leq2 \int_{B_{R_{1}}}\frac{\rho(x)}{R_{1}}\,dx \int_{B_{R_{2},\infty}}\frac{\rho(y)}{|y|^{2}}\,dy \leq\frac{C_{1}}{R_{2}}. $$

Next, we estimate \(I_{12}\) and \(I_{23}\):

$$\begin{aligned} I_{12}+ I_{23} ={}& {-} \int\rho_{1} \Phi_{2} \,dx- \int\rho_{2} \Phi_{3}\,dx = \frac{1}{4\pi g} \int\nabla(\Phi_{1} +\Phi_{3})\cdot\nabla \Phi_{2} \,dx \\ \leq{}& C_{2} \Vert \rho_{1} +\rho_{3} \Vert _{\frac{6}{5}} \Vert \nabla\Phi_{2} \Vert _{2} \leq C_{3} \Vert \nabla\Phi_{2} \Vert _{2}. \end{aligned}$$

If we define \(M_{l} =\int\rho_{l} \,dx\), then it is easy to see that \(M=M_{1}+ M_{2} +M_{3}\).

The remaining proofs are carried out in the same way as for Theorem 2.1 in [1], except that instead of the erroneous expressions (2.4) and (2.8), we have to use their corrected versions given in Sect. 2. □