A two-phase problem with a lower-dimensional free boundary
type: Item
instance of: scholarly article
title: A two-phase problem with a lower-dimensional free boundary
zbMATH Open document ID: 1260.35256
DOI: 10.4171/IFB/283
author: Arshak Petrosyan
author: Mark G. Allen
published in: Interfaces and Free Boundaries
publication date: 2013-01-17T00:00:00Z
review text: Summary: For a bounded domain \(D\subset \mathbb R^n\), we study minimizers of the energy functional \[ \int_{D}|\nabla u|^2\,dx + \int_{D \cap (\mathbb R^{n-1} \times \{0\} )} \lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u < 0\}} \,d\mathcal{H}^{n-1}, \] without any restriction on the function \(u\). One of the main results states that the free boundaries \[ \Gamma^+ = \partial \{u(\cdot, 0) > 0\}\quad \text{and}\quad \Gamma^- = \partial \{u(\cdot, 0) < 0\} \] never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension \(n=3\) the free boundaries are \(C^1\) regular on a dense subset.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 6126404
zbMATH Keywords: two-phase free boundary problem
zbMATH Keywords: lower-dimensional free boundary
zbMATH Keywords: separation of phases
zbMATH Keywords: regularity of the free boundary
zbMATH Keywords: monotonicity formula
zbMATH Keywords: Alexandrov reflection technique
zbMATH Keywords: Steiner symmetrization
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/ifb/283
OpenAlex ID: W1963977425
description: scientific article; zbMATH DE number 6126404