We study the regularity of minimizers for a variant of the soap bubble cluster problem: \(\begin{aligned} \min \sum _{\ell =0}^N c_{\ell } P( S_\ell )\,, \end{aligned}\) where \(c_\ell >0\) , among partitions \(\{S_0,\dots ,S_N,G\}\) of \(\mathbb {R}^2\) satisfying \(|G|\le \delta \) and an area constraint on each \(S_\ell \) for \(1\le \ell \le N\) . If \(\delta >0\) , we prove that for any minimizer, each \(\partial S_{\ell }\) is \(C^{1,1}\) and consists of finitely many curves of constant curvature. Any such curve contained in \(\partial S_{\ell } \cap \partial S_{m}\) or \(\partial S_\ell \cap \partial G\) can only terminate at a point in \(\partial G \cap \partial S_\ell \cap \partial S_{m}\) at which G has a cusp. We also analyze a similar problem on the unit ball B with a trace constraint instead of an area constraint and obtain analogous regularity up to \(\partial B\) . Finally, in the case of equal coefficients \(c_\ell \) , we completely characterize minimizers on the ball for small \(\delta \) : they are perturbations of minimizers for \(\delta =0\) in which the triple junction singularities, including those possibly on \(\partial B\) , are “wetted”by G.
