Conti–Otto–Serfaty dimension conjecture for the irrigated measure

Consider the branched transport energy functional referred to as

, and let $\mu$ be a minimizer. Denote by $\mu_{\pm T}$ its trace at the boundary \times $\pm T$. The **Conti–Otto–Serfaty dimension conjecture.** If $\mu$ is a minimizer of

, then $\mu_{\pm T}$ is of dimension $8/5$. This conjecture predicts the Hausdorff dimension of the boundary trace in the crossover regime of the branched transport model for type-I superconductors. The paper explains that proving it is equivalent to establishing local energy bounds with an optimal exponent; the bounds obtained there are not optimal.