Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with $L^1$-type deviations

The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in [G. Ciraolo and F. Maggi, Comm. Pure Appl. Math. (2017)], where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in [V. Julin and J. Niinikoski, Anal. PDE (2023)], where $L^{N-1}$-deviations are considered for domains in $\mathbb{R}^N$. In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation $$ \int_{\partial \Omega } \left( H_0 - H \right)^+ dS_x, \quad \text{ where } \begin{cases} H \text{ is the mean curvature of } \partial \Omega , \\ H_0:=\frac{| \partial \Omega |}{N | \Omega |} , \\ \left( H_0 - H \right)^+:=\max\left\lbrace H_0 - H , 0 \right\rbrace , \end{cases} $$ which is clearly even weaker than $\Vert H_0-H \Vert_{L^1( \partial \Omega )}$.