In this talk, we explore perturbed isoperimetric problems, focusing on the minimization of an energy functional composed of a perimeter term P and an additional perturbation term V. Specifically, we aim to identify a set of conditions under which the problem admits "generalized minimizers" with density estimates. Our framework accommodates various forms of P, including the classical distributional perimeter, anisotropic perimeter, and fractional perimeter. The perturbation term V can take the form of a Riesz kernel, an integral involving a periodic function, or a spectral functional.
Idriss Mazari
