An overview of unconstrained free boundary problems

Abstract

In this paper, we present a survey concerning unconstrained free boundary problems of typewhere B 1 is the unit ball, Ω is an unknown open set, F 1 and F 2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

Free boundary limit of a tumor growth model with nutrient

“…The Hele-Shaw problem is a widely studied free boundary model. Although we are only interested in the weak formulation, the regularity of the boundary is also a challenging issue, see [10,16,26].…”

Free boundary problems: the forefront of current and future developments

“…A. Figalli and H. Shahgholain in [21] present a survey on unconstrained free boundary problems involving elliptic operators. The main objective is to discuss a unifying approach to the optimal regularity of solutions to these matching problems.…”

Analysis of blow-ups for the double obstacle problem in dimension two

“…It has been shown that the solution to the double obstacle problem is locally C 1,1 under the assumption ψ i ∈ C 2 (Ω), see for instance [3,5]. Therefore we may rewrite (1.2) as ψ 1 ≤ u ≤ ψ 2 and ∆u = ∆ψ 1 χ {u=ψ 1 } + ∆ψ 2 χ {u=ψ 2 } − ∆ψ 1 χ {ψ 1 =ψ 2 } a.e., (1.3) where χ A is the characteristic function of a set A ⊂ R n .…”