Null Controllability of the Heat Equation with Mixed Boundary Conditions in a Cracked Domain

The objective of this work is the study of the controllability of the heat equation subject to mixed boundary conditions in a cracked domain. The solution of the problem may be singular near the tips of the crack and near the points where Neumann and Dirichlet boundary conditions meet. So, in this work, we combine two types of singularities (mixed boundary conditions and nonsmooth domains). This analysis is based on Carleman estimates and the crucial point is to adapt these estimates to our problem in the presence of singularities, by choosing a suitable weight function. To our knowledge, the weight function usually has the same sign throughout the domain, but the existence of such function is not evident in our case. So, we need to construct it differently.