Regularity of the Free Boundary for Measure Constrained Minimizers
Bozhidar Velichkov · Lecture notes of the Unione Matematica Italiana · 2022
Abstract Let D be a connected bounded open set in $$\mathbb {R}^d$$ ℝ d and let v ∈ H1(D) be a given non-negative function. This chapter is dedicated to the problem $$\displaystyle \begin{aligned} \min\Big\{\mathcal F_0(u,D)\ :\ u\in H^1(D),\ u-v\in H^1_0(D),\ |\Omega_u\cap D|=m\Big\}, \end{aligned} $$ min { ℱ 0 ( u , D ) : u ∈ H 1 ( D ) , u − v ∈ H 0 1 ( D ) , | Ω u ∩ D | = m } , where m ∈ (0, |D|) is a fixed constant and we recall that $$\displaystyle \begin{aligned} \mathcal F_0(u,D)=\int_D| abla u|{}^2\,dx. \end{aligned}$$ ℱ 0 ( u , D ) = ∫ D | ∇ u | 2 d x . In this chapter, we give the main steps of the proof of Theorem 1.9 .