SwissMAP Logo
Log in
  • About us
    • Organization
    • Professors
    • Senior Researchers
    • Postdocs
    • PhD Students
    • Alumni
  • News & Events
    • News
    • Events
    • Online Events
    • Videos
    • Newsletters
    • Press Coverage
    • Perspectives Journal
    • Interviews
  • Research
    • Basic Notions
    • Phase III Directions
    • Phases I & II Projects
    • Publications
    • SwissMAP Research Station
  • Awards, Visitors & Vacancies
    • Awards
    • Innovator Prize
    • Visitors
    • Vacancies
  • Outreach & Education
    • Masterclasses & Doctoral Schools
    • Mathscope
    • Maths Club
    • Athena Project
    • ETH Math Youth Academy
    • SPRING
    • Junior Euler Society
    • General Relativity for High School Students
    • Outreach Resources
    • Exhibitions
    • Previous Programs
    • Events in Outreach
    • News in Outreach
  • Equal Opportunities
    • Mentoring Program
    • Financial Support
    • SwissMAP Scholars
    • Events in Equal Opportunities
    • News in Equal Opportunities
  • Contact
    • Corporate Design
  • Basic Notions
  • Phase III Directions
  • Phases I & II Projects
  • Publications
  • SwissMAP Research Station

Constraint maps with free boundaries: the Bernoulli case

Alessio Figalli, André Guerra, Sunghan Kim, Henrik Shahgholian

7/11/23 Published in : arXiv:2311.03006

In this manuscript, we delve into the study of maps that minimize the Alt-Caffarelli energy functional \int_\Omega (|Du|^2 + q^2 \chi_{u^{-1}(M)})\,dx,, under the condition that the image u(\Omega) is confined within \overline M. Here, Ω denotes a bounded domain in the ambient space \mathbb{R}^n (with n≥1), and M represents a smooth domain in the target space \mathbb{R}^m (where m≥2). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, {\rm int}(u^{-1}(\partial M)), such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary. Our first significant contribution is the validity of a ε-regularity theorem. This theorem is founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small energy. Our subsequent key finding reveals that, whenever the complement of M is uniformly convex and of class C^3, the maps minimizing the Alt-Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the non-coincidence set u^{-1}(M). In particular, this Lipschitz continuity extends to the free boundary. A noteworthy consequence of our findings is the smoothness of flat free boundaries and of the resulting image maps

Entire article

Phase I & II research project(s)

  • Statistical Mechanics

Phase III direction(s)

  • Differential equations of Mathematical Physics

Sharp quantitative stability of the Brunn-Minkowski inequality

Black hole bulk-cone singularities

  • Leading house

  • Co-leading house


The National Centres of Competence in Research (NCCRs) are a funding scheme of the Swiss National Science Foundation

© SwissMAP 2025 - All rights reserved