An overview of unconstrained free boundary problems

2018/05/24 by Alessio Figalli, Henrik Shahgholian, Figalli, Alessio +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1805.09726

openalex publication_date 2018/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we present a survey concerning unconstrained free boundary problems of type \ F₁(D²u,∇ u,u,x)=0 · amp; in B₁ ∩ Ω,
F₂ (D² u,∇ u,u,x)=0 · amp; in B₁∖Ω,
u ∈ \mathbbS(B₁), . where B₁ is the unit ball, Ω is an unknown open set, F₁, F₂ are elliptic operators (admitting regular solutions), and \mathbbS 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.

Citations

• C^1,α regularity of solutions of degenerate fully non-linear elliptic equations
• On the Equality of Solutions of Max-Min and Min-Max Systems of Variational Inequalities with Interconnected Bilateral Obstacles
• Normal random matrix ensemble as a growth problem

Cited by

• Homogenization of a vertical oscillating Neumann condition
• Lagrange multipliers and characteristic functions

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