Suppose $G \subset \mathbb{C}$ is a bounded domain, such that $\partial G$ has positive capacity, and let $v$ be a unit mass measure compactly supported in $G$. Then the balayage problem is to find a measure $\hat{v}$ supported on $\partial G$, of unit mass, and of bounded logarithmic potential $U^{\hat{v}}$ on $\partial G$, satisfying $U^{\hat{v}} = U^{v}$ quasi-everywhere on $\partial G$, and everywhere outside $\overline{G}$.
One way to solve the balayage problem, is to solve the Dirichlet problem in $G$ with boundary values given by $U^{v}$, and then take the lower-semicontinuous regularisation. That is, let $H_{U^{v}}$ be the Perron solution to this Dirichlet problem in $G$, (i.e. the upper-envelope of all subharmonic functions with $\limsup_{z \rightarrow \zeta} u(z) \leq U^{v}(\zeta)$ for all $\zeta \in \partial G$), and set $V$ equal to $H_{U^{v}}$ in $G$, $U^{v}$ outside $\overline{G}$, and $\liminf_{z \rightarrow \zeta} H_{U^{v}}(z)$ at boundary points $\zeta \in \partial G$.
Then $V(\zeta) = U^{v}(\zeta)$ at quasi-every boundary point $\zeta \in \partial G$, in particular at every regular boundary point of $G$ this holds (since $U^{v}$ is continuous on $\partial G$, as $v$ is compactly supported in $G$)
It should be the case that $V$ is the potential of a measure supported on $\partial G$, but by the Riesz decomposition theorem, I can only recover $V = U^{-\frac{1}{2 \pi} \Delta V} + h$ on relatively compact domains containing $\partial G$,where $h$ is harmonic, and do not know how to show $h$ must in fact be equal to zero.