In this paper we state a hypoellipticity result in the framework of microlocal boundary value problems (which has to be compared with the analogous results of [SKK], [KS2] at the interior). More precisely, let $\mathcal {M}$ be a system of microdifferential equations with simple characteristics on a complex manifold $X$, and let $\Lambda _i$ ($i=1,2$) be a pair of real Lagrangian submanifolds of $T^*X$. Denote by $\mathcal {C}_{\Lambda _i}$ the associated complexes of microfunctions. If the pair $(\Lambda _1,\Lambda _2)$ is « positive », we prove the injectivity of the natural « restriction »morphism $\mathop {\mathcal {E}xt}\nolimits ^j_{\mathcal {E}_X}(\mathcal {M},\mathcal {C}_{\Lambda _2})\longrightarrow \mathop {\mathcal {E}xt}\nolimits ^j_{\mathcal {E}_X}(\mathcal {M},\mathcal {C}_{\Lambda _1})$ between solution sheaves, where $j$ is the first possibly non-vanishing degree of the cohomology.