Quantitative isoperimetric inequalities for classical capillarity problems

We consider capillarity functionals which measure the perimeter of sets contained in a Euclidean half-space assigning a constant weight $λ\in (-1,1)$ to the portion of the boundary that touches the boundary of the half-space. Depending on $λ$, sets that minimize this capillarity perimeter among those with fixed volume are known to be suitable truncated balls lying on the boundary of the half-space. We first give a new proof based on an ABP-type technique of the sharp isoperimetric inequality f...