Logarithmic capacity and the two-dimensional Laplace layer kernel

For $x\mathrm{\ne }y$ in ${\mathbb{R}}^2$, the logarithmic Laplace fundamental solution is $G_2(x,y)=-(2\pi {)}^{-1}\log \mathrm{\mid }x-y\mathrm{\mid }$. If $\mathrm{\Gamma }$ is a closed piecewise-smooth curve and $\lambda$ is a smooth density, its two-dimensional single-layer trace is $V_{\mathrm{\Gamma }}\lambda (x)=-\frac{1}{2\pi }{\int }_{\mathrm{\Gamma }}\log \mathrm{\mid }x-y\mathrm{\mid }\lambda (y)d{s}_y\mathrm{.}$

For a compact set $K\subset {\mathbb{R}}^2$ with positive finite logarithmic capacity, let $\mathcal{P}(K)$ be the Borel probability measures supported on $K$, let $I(\mu )={\int }_K{\int }_K\log \frac{1}{\mathrm{\mid }x-y\mathrm{\mid }}d\mu (x)d\mu (y)$ with value $+\mathrm{\infty }$ on the diagonal, and define $I(K)={\inf }_{\mu \in \mathcal{P}(K)}I(\mu ),\mathrm{cap}(K)=e^{-I(K)}\mathrm{.}$ This definition fixes length units: scaling $K$ changes its capacity and adds a constant-mode term to the logarithmic layer operator, as proved separately by the homothety theorem on this page.