TY - JOUR

T1 - Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms

AU - Breit, Dominic

AU - Cianchi, Andrea

N1 - Funding Information:
Research Project 201758MTR2 of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 ?Direct and inverse problems for partial differential equations: theoretical aspects and applications?;GNAMPA of the Italian INdAM ? National Institute of High Mathematics (grant number not available).
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/11/19

Y1 - 2021/11/19

N2 - A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

AB - A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

KW - Extension operator

KW - K-functional

KW - Orlicz-Sobolev spaces

KW - Rearrangement-invariant norms

KW - Sobolev inequality

KW - Symmetric gradient

UR - http://www.scopus.com/inward/record.url?scp=85113361120&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107954

DO - 10.1016/j.aim.2021.107954

M3 - Article

SN - 0001-8708

VL - 391

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107954

ER -