To the theory of variational method for Beltrami equations
type: Item
instance of: scholarly article
title: To the theory of variational method for Beltrami equations
zbMATH Open document ID: 1283.30050
author: Vladimir Gutlyanskiĭ
author: T. V. Lomako
author: Vladimir I. Ryazanov
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-28T00:00:00Z
review text: The authors extend the variational method for the Beltrami equation \(f_{\bar z}=\mu f_z\) (\(\|\mu\|_\infty=k<1\)) to the setting of degenerate Beltrami equations, which allows for \(\|\mu\|_\infty=1\). In the non-degenerate case it is well-known that extremal maps tend to have constant value of \(|\mu|\), which denotes the modulus of the complex dilatation. The degenerate case is quite different because the constraint on \(\mu\) may have an integral form, e.g., \(\int_{\mathbb C} \Psi(|\mu|)\leq 1\), instead of pointwise \(|\mu|\leq k\) a.e. In particular, one cannot expect extremal maps to have constant \(|\mu|\). The authors prove that for a wide range of variational problems, any extremal map will attain equality in the integral dilatation constraint \(\int_{\mathbb C} \Psi(|\mu|) = 1\). Also, if the constraint has the form of a differential inclusion, \(\mu(z)\in M(z)\subset \{z:|z|<1\}\), then an extremal map satisfies \(\mu(z)\in \partial M(z)\) almost everywhere.
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30C70
zbMATH DE Number: 6110724
zbMATH Keywords: degenerate Beltrami equation
zbMATH Keywords: variational method
zbMATH Keywords: extremal homeomorphisms
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
full work available at URL: s10958-012-0727-1
OpenAlex ID: W2059201541
cites work: Q5512337
cites work: Riemann's mapping theorem for variable metrics
cites work: On the order of closeness of a spatial quasiconformal mapping to conformal
cites work: On the Beltrami equations with two characteristics
cites work: On integral conditions for the general Beltrami equations
cites work: Q4825139
cites work: Solutions de l'equation de Beltrami avec ||\mu||_\infty = 1
cites work: Q5565773
cites work: Q5574105
cites work: Q3719272
cites work: Boundary behavior of quasiisometric mappings
cites work: The product of conformal radii of nonoverlapping domains
cites work: Regularity of the inverse of a planar Sobolev homeomorphism
cites work: Q5650526
cites work: Q5685358
cites work: On the theory of convergence and compactness for Beltrami equations
cites work: Moduli in Modern Mapping Theory
cites work: The N\(^{-1}\)-property of maps and Luzin's condition (N)
cites work: Q5567211
cites work: Some questions of convergence and compactness for quasiconformal mappings
cites work: Quasiconformal mappings with restrictions in measure
cites work: On strong solutions of the Beltrami equations
cites work: Solution of one boundary value problem for curl, +λ I operator in the ball
cites work: A variational method for general families of quasiconformal mappings
cites work: On mappings generating the embeddings of Sobolev spaces
cites work: A classification of sub-Riemannian manifolds
DOI: 10.1007/S10958-012-0727-1
description: scientific article; zbMATH DE number 6110724