Quantitative stability of extremal quasi conformal mappings
Zolt\'an M. Balogh, K\'aroly J. B\"or\"oczky, \'Agnes Mester
TL;DR
This paper proves that mappings with near-minimal mean distortion are quantitatively close to extremal solutions, providing sharp stability estimates in the theory of quasiconformal mappings.
Contribution
It establishes the first quantitative stability results for classical distortion minimization problems in quasiconformal mapping theory.
Findings
Sharp stability estimates for extremal maps
Quantitative closeness in Lebesgue norms
Stability results for mean distortion minimization
Abstract
We establish quantitative stability results for classical distortion minimization problems in the theory of quasiconformal mappings. We consider the mean distortion functional and prove sharp stability estimates for the minimization problems regarding the linear stretch and spiral stretch maps, which arise as extremals in the class of mappins with finite distortion under appropriate boundary conditions. More precisely, we show that if a mapping has mean distortion close to the minimal value in the appropriate function class, then it must be quantitatively close, in certain Lebesgue norms.
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Taxonomy
TopicsAnalytic and geometric function theory · Stability and Controllability of Differential Equations · Geometry and complex manifolds