We consider the question of describing the envelope of holomorphy of a relatively open part of the boundary of a strongly pseudoconvex domain. In the two-dimensional case a complete answer to this question is known ; moreover it is known what are, in every dimension $n \geq 2$, the necessary and sufficient conditions for the envelope to be the whole domain. Here we prove a theorem that establishes, for general $n \geq 2$, the necessary and sufficient conditions for the envelope to be the complement of a given compact set. The theorem generalizes the previously known results and provides further information on this subject.