This thesis achieves a theoretical and numerical studies of cavities partially filled with a negative material, that is a material for which the magnetic permeability and/or the electric permittivity (or at least their real part) become negative in some frequency ranges. This study is part of the main thrust of the work started in our team focusing on the electromagnetic wave propagation in presence of such negative materials, at a given fraquency. The purpose of this thesis is to take into account the frequency dispersion, that is the frequency dependence of the permeability and/or the permittivity, considering the frequency as the spectral parameter. We highlight the essential spectrum arising from the presence of negative material, as well as the resulting resonance phenomena, for different models describing this material. The theoretical study focuses on the case of polygonal bi-dimensional cavities for the Drude and the Lorentz models (with dissipation or not). The theoretical study of the simplest model (the non dissipative Drude model) is extended to the case of a curved (but regular) interface. This model is also the subject of a numerical study, aimed at exploring the effect of a finite element discretization of the theoretical problem, and thus highlight the difficulties to numerically notice some of the resonance phenomena.