9 sep 2021 -- 14:50 [open in google calendar]
It is well-known that the smallest p-capacity among sets of a given volume is achieved solely by balls. A natural question to ask is whether the balls are stable minimizers. That is, supposing a set has almost the p-capacity of a ball with the same volume, is it quantitatively close to it? The first result to our knowledge was given by Fusco, Maggi, and Pratelli in 2009. However, the estimate they provide is not sharp. In this talk we discuss the sharp quantitative stability of the balls for p-capacity. The proof combines the Selection Principle, introduced by Cicalese and Leonardi in 2012, with the regularity theory for free boundary problems established by Danielli and Petrosyan in 2005. This is a generalization of a joint work with Guido De Philippis and Michele Marini, where we got the sharp isocapacitary inequaity for p = 2.