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September 5, 2025Advances in Calculus of Variations

A Sobolev gradient flow for the area-normalised Dirichlet energy of H 1 maps

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Authors

SOShinya OkabeChibune General HospitalPSPhilip SchraderMurdoch UniversityGWGlen WheelerUniversity of Wollongong

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Overview

Analysis of Sobolev gradient flow reveals eternal existence of solutions in H1 maps, suggesting improved isoperimetric inequality.

Key Points

  • Solutions with initially positive signed enclosed area exist eternally and converge to a circle.
  • The evolving area-normalised Dirichlet energy showcases the relationship between shape and area.
  • The study enhances understanding of map dynamics in Sobolev spaces, particularly regarding isoperimetric inequalities.
  • Each circle attained through the gradient flow may be multiply-covered, suggesting multiple solutions.

Cite This Study

Okabe et al. (2025) studied this question.

synapsesocial.com/papers/68bb421a2b87ece8dc958902https://doi.org/10.1515/acv-2024-0081
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