Upper large deviations for the maximal flow through a domain of bolds{mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$ representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behavior of the flow $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between the corresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of $\phi_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of $\phi_n/n^{d-1}$.