Analytic characterization of monotone Hopf-harmonics

Ilmari Kangasniemi, Aleksis Koski, Jani Onninen

Research output: Contribution to journalArticlepeer-review


We study solutions of the inner-variational equation associated with the Dirichlet energy in the plane, given homeomorphic Sobolev boundary data. We prove that such a solution is monotone if and only if its Jacobian determinant does not change sign. These solutions, called monotone Hopf-harmonics, are a natural alternative to harmonic homeomorphisms. Examining the topological behavior of a solution (not a priori monotone) on the trajectories of Hopf quadratic differentials plays a sizable role in our arguments.

Original languageEnglish (US)
Article number140
JournalCalculus of Variations and Partial Differential Equations
Issue number4
StatePublished - Aug 2022

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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