Approximation in Sobolev spaces of nonlinear expressions involving the gradient
type: Item
instance of: scholarly article
title: Approximation in Sobolev spaces of nonlinear expressions involving the gradient
zbMATH Open document ID: 1029.46028
DOI: 10.1007/BF02384536
author: Piotr Hajłasz
author: Jan Malý
published in: Arkiv för Matematik
publication date: 2003-09-30T00:00:00Z
review text: The aim of the paper under review is to approximate mappings \(u:\Omega\rightarrow {\mathbb{R}}^m\) from a Sobolev space \(W^{1,p}(\Omega)^m\) (\(1\leq p<\infty\)) on an open set \(\Omega\subset {\mathbb{R}}^n\) by a sequence \(\{ u_k\}_{k=1}^\infty\) of mappings with some good regularity conditions. In the main theorem it is shown that the approximating functions \(u_k\) can be taken such that \(\nabla u_k=0\) a.e. in the set where \(u_k\neq u\), and \(u_k\) locally weakly monotone in the sense of \textit{J. J. Mamfredi} [J. Geom. Anal. 4, 393-402 (1994; Zbl 0805.35013)]. This allows the authors to obtain the convergence \(\int_\Omega f(x,u_k,\nabla u_k) dx \to\int_\Omega f(x,u,\nabla u) dx\) for a large class of integrands, which include many interesting special cases, such as \(\det \nabla u_k \to \det \nabla u\) in \(L^q(\Omega)\). When \(p=n\), every mapping \(u_k\) is continuous, differentiable a.e. and satisfies the Lusin property (the image of any set of Lebesgue measure zero is also of Lebesgue measure zero), and it is proved that this result is sharp.
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1987657
zbMATH Keywords: Sobolev space
zbMATH Keywords: approximating functions
reviewed by: Joan Cerdà
MaRDI profile type: Publication
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Global invertibility of Sobolev functions and the interpenetration of matter
cites work: Q3689532
cites work: Q4321362
cites work: Analytical foundations of the theory of quasiconformal mappings in R^n
cites work: Q5800508
cites work: Direct methods in the calculus of variations
cites work: A direct variational approach to Skyrme's model for meson fields
cites work: Sobolev maps with integer degree and applications to Skyrme’s problem
cites work: Q5565773
cites work: Q4502667
cites work: Q3198418
cites work: Graphs of finite mass which cannot be approximated in area by smooth graphs
cites work: Q4210546
cites work: Change of variables formula under minimal assumptions
cites work: A note on weak approximation of minors
cites work: Sobolev spaces on an arbitrary metric space
cites work: Hölder quasicontinuity of Sobolev functions on metric spaces
cites work: On the differentiability of solutions of quasilinear elliptic equations
cites work: Sobolev mappings with integrable dilatations
cites work: Mappings of finite distortion: Monotonicity and continuity
cites work: Q4299111
cites work: Riemannian geometry and geometric analysis.
cites work: Q4344980
cites work: Regularity theory and traces of 𝒜-harmonic functions
cites work: Q4139197
cites work: Q4146111
cites work: Q3989702
cites work: Hölder type quasicontinuity
cites work: Absolutely continuous functions of several variables
cites work: A simple proof of the Stepanov theorem on differentiability almost everywhere
cites work: Lusin's condition (N) and mappings of the class W1, n.
cites work: Weakly monotone functions
cites work: Traces of monotone Sobolev functions
cites work: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems
cites work: Lusin's condition (N) and mappings with nonnegative Jacobians
cites work: Banach lattices
cites work: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms
cites work: Approximation in area of graphs with isolated singularities
cites work: Approximation in area of continuous graphs
cites work: Graphs of finite mass which cannot be approximated by smooth graphs with equibounded area
cites work: An existence theory for nonlinear elasticity that allows for cavitation
cites work: Q5607484
cites work: Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients
cites work: Pointwise differentiability properties of solutions of quasilinear parabolic equations
cites work: Regularity properties of deformations with finite energy
cites work: Infima of energy functionals in homotopy classes of mappings
cites work: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps
Recommended article: Q3528053
Recommended article: Q3029587
Recommended article: Q4492804
Recommended article: Approximation of sobolev mappings
Recommended article: Q3805279
Recommended article: Sobolev gradients and differential equations
Recommended article: Sobolev gradients in uniformly convex spaces
Recommended article: Q4953899
Recommended article: Q3985301
description: scientific article; zbMATH DE number 1987657