Proof of the remaining cases of the Yamabe boundary problem

In this paper, we solve the remaining cases of the boundary Yamabe problem introduced by Escobar in 1992. Indeed, using the bubbles of Brendle-Chen, which are an adaptation to manifolds with boundary of the original ones introduced by Brendle for the study of the Yamabe flow on closed Riemannian manifolds of dimension greater or equal to 6, and the algebraic topological argument of Bahri-Coron, we solve the cases left open after the work of Brendle-Chen. Thus combining our work with the ones of Brendle-Chen and Escobar, we have that the boundary Yamabe problem is a done deal.