The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

Lyonell Boulton; Jan Lang

doi:10.1016/j.na.2023.113362

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

Lyonell Boulton^*
, Jan Lang

^*Corresponding author for this work

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

38 Downloads (Pure)

Abstract

We consider the non-linear eigenvalue equations characterizing L^p into L^q Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)². Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if [Formula presented]. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

Original language	English
Article number	113362
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	236
Early online date	22 Aug 2023
DOIs	https://doi.org/10.1016/j.na.2023.113362
Publication status	Published - Nov 2023

Keywords

Higher order Sobolev embeddings
pq-biLaplacian
s-numbers

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.na.2023.113362Licence: CC BY

Download pdf
Crown Copyright © 2023.
Final published version, 799 KBLicence: CC BY

Cite this

@article{f209433a465e419db9475e6638ecc632,

title = "The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings",

abstract = "We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if [Formula presented]. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.",

keywords = "Higher order Sobolev embeddings, pq-biLaplacian, s-numbers",

author = "Lyonell Boulton and Jan Lang",

note = "Funding Information: This research was funded by The UK's Royal Society International Exchange Grant “James Orthogonality and Higher order Sobolev Embeddings”. We are also grateful to our colleagues at the Czech Technical University in Prague for hosting many of the discussions that eventually lead to this paper. Jan Lang was partially supported by OP RDE project No. CZ.02.2.69/0.0/0.0/18\_053/0016976 International mobility of research, technical and administrative staff at the Charles University. Funding Information: This research was funded by The UK{\textquoteright}s Royal Society International Exchange Grant “James Orthogonality and Higher order Sobolev Embeddings”. We are also grateful to our colleagues at the Czech Technical University in Prague for hosting many of the discussions that eventually lead to this paper. Jan Lang was partially supported by OP RDE project No. CZ.02.2.69/0.0/0.0/18\_053/0016976 International mobility of research, technical and administrative staff at the Charles University. Publisher Copyright: {\textcopyright} 2023",

year = "2023",

month = nov,

doi = "10.1016/j.na.2023.113362",

language = "English",

volume = "236",

journal = "Nonlinear Analysis, Theory, Methods and Applications",

issn = "0362-546X",

publisher = "Elsevier Ltd",

}

TY - JOUR

T1 - The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

AU - Boulton, Lyonell

AU - Lang, Jan

N1 - Funding Information: This research was funded by The UK's Royal Society International Exchange Grant “James Orthogonality and Higher order Sobolev Embeddings”. We are also grateful to our colleagues at the Czech Technical University in Prague for hosting many of the discussions that eventually lead to this paper. Jan Lang was partially supported by OP RDE project No. CZ.02.2.69/0.0/0.0/18_053/0016976 International mobility of research, technical and administrative staff at the Charles University. Funding Information: This research was funded by The UK’s Royal Society International Exchange Grant “James Orthogonality and Higher order Sobolev Embeddings”. We are also grateful to our colleagues at the Czech Technical University in Prague for hosting many of the discussions that eventually lead to this paper. Jan Lang was partially supported by OP RDE project No. CZ.02.2.69/0.0/0.0/18_053/0016976 International mobility of research, technical and administrative staff at the Charles University. Publisher Copyright: © 2023

PY - 2023/11

Y1 - 2023/11

N2 - We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if [Formula presented]. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

AB - We consider the non-linear eigenvalue equations characterizing Lp into Lq Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the extremal functions in the general case (p,q)∈(1,∞)2. Among other results, we show that these can be expressed in terms of those of related first order embeddings, if and only if [Formula presented]. Our findings shed new light on the surprising nature of higher order Sobolev spaces in the Banach space setting.

KW - Higher order Sobolev embeddings

KW - pq-biLaplacian

KW - s-numbers

UR - https://www.scopus.com/pages/publications/85168558412

U2 - 10.1016/j.na.2023.113362

DO - 10.1016/j.na.2023.113362

M3 - Article

AN - SCOPUS:85168558412

SN - 0362-546X

VL - 236

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

M1 - 113362

ER -

The eigenvalues and eigenfunctions of the non-linear equation associated to second order Sobolev embeddings

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this