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 lambda is an element of(-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in (-1,1)$$\end{document} to the portion of the boundary that touches the boundary of the half-space. Depending on lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, 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 for this class of capillarity problems. Next we prove two quantitative versions of the inequality: a first sharp inequality estimates the Fraenkel asymmetry of a competitor with respect to the optimal bubbles in terms of the energy deficit; a second inequality estimates a notion of asymmetry for the part of the boundary of a competitor that touches the boundary of the half-space in terms of the energy deficit. After a symmetrization procedure, the quantitative inequalities follow from a novel combination of a quantitative ABP method with a selection-type argument.