Dissertation: Understanding mappings and values of finite distortion (Zhu)

Doctoral Researcher Yizhe Zhu studies mappings of finite distortion, a class of functions that plays an important role in geometric function theory and nonlinear elasticity. Such mappings describe how space can be deformed when stretching is allowed to vary from point to point, while still preserving meaningful geometric information.

- Mappings of finite distortion and values of finite distortion can be viewed as a flexible generalization of quasiregular mappings. They arise naturally both in pure mathematics and in models of elastic materials, explains Doctoral Researcher Yizhe Zhu from the University of Jyväskylä.

Uniqueness of energy-minimizing deformations

In mathematics and elasticity, minimizers usually represent the most efficient or physically natural deformation between two regions. Zhu studied conditions that guarantee the existence and uniqueness of such diffeomorphic minimizers.

- Existence of such minimizers is still a difficult open problem in general, but we can prove such a diffeomorphic minimizer is unique if it exists, says Zhu.

The Jacobian determinant: understanding local volume change

The thesis also studies the Jacobian determinant, which measures how a mapping changes volume locally. Understanding when the reciprocal of the Jacobian is integrable is important, since it controls how much a mapping can compress volume.

- This result connects analytic estimates with the geometric and physical behavior of deformations, Zhu explains.

A generalized classes under scrutiny

Zhu also studied related generalized classes of mappings, investigating when such mappings preserve sets of measure zero and when a single-value version of Reshetnyak’s theorem remains valid. 

- The classical Reshetnyak’s theorem and the Lusin (N) -property are fundamental results in quasiregular mapping theory. Extending these results to such generalized classes of mappings is an interesting and natural generalization, says Zhu.

M.Sc Yizhe Zhu defends his doctoral dissertation "Mappings of finite distortion and values of finite distortion: uniqueness of diffeomorphic minimizers, global integrability of the Jacobian, Reshetnyak's theorem” on 8.6.2026 at 12:00 at Agora Auditorio 3 (Ag B103). Opponent is University Lecture Sari Rogovin and custos is Professor Tapio Rajala (University of Jyväskylä). The language of the defense is English.

The dissertation is available at https://urn.fi/URN:ISBN:978-952-86-1580-4