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

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}}^d\mathrm{/}n$ for $d\ge 2$ and a domain $\mathrm{\Omega }$ of boundary $\mathrm{\Gamma }$ in ${\mathbb{R}}^d$. Let ${\mathrm{\Gamma }}^1$ and ${\mathrm{\Gamma }}^2$ be two disjoint open subsets of $\mathrm{\Gamma }$ representing the parts of $\mathrm{\Gamma }$ through which some water can enter and escape from $\mathrm{\Omega }$. We investigate the asymptotic behavior of the flow ${\varphi }_n$ through a discrete version ${\mathrm{\Omega }}_n$ of $\mathrm{\Omega }$ between the corresponding discrete sets ${\mathrm{\Gamma }}_n^1$ and ${\mathrm{\Gamma }}_n^2$. 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 ${\varphi }_n\mathrm{/}{n}^{d-1}$ 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 ${\varphi }_n\mathrm{/}{n}^{d-1}$.