In the present paper, lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by \textit{D. Chenais} [J. Math. Anal. Appl. 52, 189-219 (1975; Zbl 0317.49005)] is employed. For this class of domains, a basic lemma is proved which plays an essential role in the derivations of the lower- semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.