LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS

Kari Astala; Daniel Faraco; André Guerra; Aleksis Koski; Jan Kristensen

doi:10.3934/cpaa.2024069

LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS

Kari Astala, Daniel Faraco^*, André Guerra, Aleksis Koski, Jan Kristensen

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.

Original language	English
Pages (from-to)	1608-1645
Number of pages	38
Journal	Communications on Pure and Applied Analysis
Volume	23
Issue number	10
DOIs	https://doi.org/10.3934/cpaa.2024069
Publication status	Published - Oct 2024
MoE publication type	A1 Journal article-refereed

Keywords

gradient Young measure
isotropic material
lower semicontinuity
mappings of integrable distortion
nonlinear elasticity
Quasiconvexity
quasiregular maps

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10.3934/cpaa.2024069

https://ora.ox.ac.uk/objects/uuid:6ec2c67a-c65e-4466-8375-799b18706f77

Cite this

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title = "LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS",

abstract = "We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and {\v S}ver{\'a}k, plays a prominent role in our approach.",

keywords = "gradient Young measure, isotropic material, lower semicontinuity, mappings of integrable distortion, nonlinear elasticity, Quasiconvexity, quasiregular maps",

author = "Kari Astala and Daniel Faraco and Andr{\'e} Guerra and Aleksis Koski and Jan Kristensen",

year = "2024",

month = oct,

doi = "10.3934/cpaa.2024069",

language = "English",

volume = "23",

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T1 - LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS

AU - Astala, Kari

AU - Faraco, Daniel

AU - Guerra, André

AU - Koski, Aleksis

AU - Kristensen, Jan

PY - 2024/10

Y1 - 2024/10

N2 - We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.

AB - We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.

KW - gradient Young measure

KW - isotropic material

KW - lower semicontinuity

KW - mappings of integrable distortion

KW - nonlinear elasticity

KW - Quasiconvexity

KW - quasiregular maps

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JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

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ER -

LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS

Abstract

Keywords

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Koski Aleksis AT: Geometriset metodit elastisuudessa

QUAMAP: Quasiconformal Methods in Analysis and Applications

Cite this

LOWER SEMICONTINUITY, STOILOW FACTORIZATION AND PRINCIPAL MAPS

Abstract

Keywords

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Projects

Koski Aleksis AT: Geometriset metodit elastisuudessa

QUAMAP: Quasiconformal Methods in Analysis and Applications

Cite this