Fix $n\ge3$. Let $F$ be a compact susbset of $\mathbb{R}^{n}$. There are two definitions of Wiener capacity.
One is $$ \mathrm{cap}\left(F\right)=\inf\left\{ \int_{\mathbb{R}^{n}}\left|\nabla u\right|^{2}dx:u\in C_{0}^{\infty}\left(\mathbb{R}^{n}\right),u\mid_{F}=1\right\} . $$ The other is $$ \mathrm{cap}\left(F\right)=\sup\left\{ \mu\left(F\right):\int_{F}\mathcal{E}\left(x-y\right)d\mu\left(y\right)\le1\quad\text{on }\mathbb{R}^{n}\setminus F\right\} , $$ where the supremum is taken over all positive finite Radon measures $\mu$ on $F$ and $\mathcal{E}$ is the fundamental solution of $-\triangle$ in $\mathbb{R}^{n}$, i.e., $$ \mathcal{E}\left(x\right)=\frac{1}{\left(n-2\right)\omega_{n}}\left|x\right|^{2-n}, $$ where $\omega_{n}$ is the area of the unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$.
According to author, two definitions are equivalent. But I cannot find any kind of proof.
The paper I read is V. Maz'ya and M. Shubin, Discreteness of spectrum and positivity criteria for Schrödinger operators, Ann. of Math. 162 (2005) 919-942.