A1 Journal article (refereed)
Analytic characterization of monotone Hopf-harmonics (2022)

Kangasniemi, I., Koski, A., & Onninen, J. (2022). Analytic characterization of monotone Hopf-harmonics. Calculus of Variations and Partial Differential Equations, 61(4), Article 140. https://doi.org/10.1007/s00526-022-02246-z

JYU authors or editors

Publication details

All authors or editorsKangasniemi, Ilmari; Koski, Aleksis; Onninen, Jani

Journal or seriesCalculus of Variations and Partial Differential Equations



Publication year2022

Publication date20/05/2022


Issue number4

Article number140

PublisherSpringer Science and Business Media LLC

Publication countryGermany

Publication languageEnglish


Publication open accessNot open

Publication channel open access

Web address of parallel published publication (pre-print)https://arxiv.org/abs/2108.00258


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.

Keywordspartial differential equationspotential theoryfunctional analysis

Free keywords31C45; 35J25; 58E20; 74B20; 46E35; Hopf-Laplace equation; holomorphic quadratic differentials; inner-variational equations; monotone mappings, orientation-preserving Sobolev mappings; the principle of non-interpenetration of matter

Contributing organizations

Ministry reportingYes

Reporting Year2022

JUFO rating2

Last updated on 2024-14-06 at 23:46