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We consider the standard first passage percolation model in the rescaledgraph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $\Omega$ of boundary $\Gamma$in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint opensubsets of $\Gamma$ representing the parts of $\Gamma$ through which some watercan enter and escape from $\Omega$. We investigate the asymptotic behavior ofthe flow $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between thecorresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. We prove thatunder some conditions on the regularity of the domain and on the law of thecapacity of the edges, the upper large deviations of $\phi_n/n^{d-1}$ above acertain 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 thatthis constant is the a.s. limit of $\phi_n/n^{d-1}$.

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