Counterexamples to maximal regularity for operators in divergence form Article Swipe
Sebastian Bechtel
,
Connor Mooney
,
Mark Veraar
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2401.05550
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2401.05550
In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$.
Related Topics
Concepts
Counterexample
Divergence (linguistics)
Mathematics
Space (punctuation)
Order (exchange)
Operator (biology)
Pure mathematics
Combinatorics
Maximal operator
Coercivity
Discrete mathematics
Mathematical analysis
Physics
Computer science
Quantum mechanics
Bounded function
Chemistry
Economics
Gene
Transcription factor
Operating system
Linguistics
Repressor
Philosophy
Finance
Biochemistry
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2401.05550
- https://arxiv.org/pdf/2401.05550
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4390833030
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4390833030Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2401.05550Digital Object Identifier
- Title
-
Counterexamples to maximal regularity for operators in divergence formWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2024Year of publication
- Publication date
-
2024-01-10Full publication date if available
- Authors
-
Sebastian Bechtel, Connor Mooney, Mark VeraarList of authors in order
- Landing page
-
https://arxiv.org/abs/2401.05550Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2401.05550Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2401.05550Direct OA link when available
- Concepts
-
Counterexample, Divergence (linguistics), Mathematics, Space (punctuation), Order (exchange), Operator (biology), Pure mathematics, Combinatorics, Maximal operator, Coercivity, Discrete mathematics, Mathematical analysis, Physics, Computer science, Quantum mechanics, Bounded function, Chemistry, Economics, Gene, Transcription factor, Operating system, Linguistics, Repressor, Philosophy, Finance, BiochemistryTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4390833030 |
|---|---|
| doi | https://doi.org/10.48550/arxiv.2401.05550 |
| ids.doi | https://doi.org/10.48550/arxiv.2401.05550 |
| ids.openalex | https://openalex.org/W4390833030 |
| fwci | |
| type | preprint |
| title | Counterexamples to maximal regularity for operators in divergence form |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T11205 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.9980000257492065 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2610 |
| topics[0].subfield.display_name | Mathematical Physics |
| topics[0].display_name | Numerical methods in inverse problems |
| topics[1].id | https://openalex.org/T12100 |
| topics[1].field.id | https://openalex.org/fields/17 |
| topics[1].field.display_name | Computer Science |
| topics[1].score | 0.9957000017166138 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1703 |
| topics[1].subfield.display_name | Computational Theory and Mathematics |
| topics[1].display_name | Advanced Mathematical Modeling in Engineering |
| topics[2].id | https://openalex.org/T11110 |
| topics[2].field.id | https://openalex.org/fields/22 |
| topics[2].field.display_name | Engineering |
| topics[2].score | 0.9945999979972839 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2207 |
| topics[2].subfield.display_name | Control and Systems Engineering |
| topics[2].display_name | Stability and Controllability of Differential Equations |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C162838799 |
| concepts[0].level | 2 |
| concepts[0].score | 0.9429291486740112 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q596077 |
| concepts[0].display_name | Counterexample |
| concepts[1].id | https://openalex.org/C207390915 |
| concepts[1].level | 2 |
| concepts[1].score | 0.8082431554794312 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q1230525 |
| concepts[1].display_name | Divergence (linguistics) |
| concepts[2].id | https://openalex.org/C33923547 |
| concepts[2].level | 0 |
| concepts[2].score | 0.756725013256073 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[2].display_name | Mathematics |
| concepts[3].id | https://openalex.org/C2778572836 |
| concepts[3].level | 2 |
| concepts[3].score | 0.669793963432312 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q380933 |
| concepts[3].display_name | Space (punctuation) |
| concepts[4].id | https://openalex.org/C182306322 |
| concepts[4].level | 2 |
| concepts[4].score | 0.6331356167793274 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q1779371 |
| concepts[4].display_name | Order (exchange) |
| concepts[5].id | https://openalex.org/C17020691 |
| concepts[5].level | 5 |
| concepts[5].score | 0.5528900623321533 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q139677 |
| concepts[5].display_name | Operator (biology) |
| concepts[6].id | https://openalex.org/C202444582 |
| concepts[6].level | 1 |
| concepts[6].score | 0.5158076286315918 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[6].display_name | Pure mathematics |
| concepts[7].id | https://openalex.org/C114614502 |
| concepts[7].level | 1 |
| concepts[7].score | 0.4393959045410156 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[7].display_name | Combinatorics |
| concepts[8].id | https://openalex.org/C3018319951 |
| concepts[8].level | 3 |
| concepts[8].score | 0.42741888761520386 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q6795639 |
| concepts[8].display_name | Maximal operator |
| concepts[9].id | https://openalex.org/C126530901 |
| concepts[9].level | 2 |
| concepts[9].score | 0.42471766471862793 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q432635 |
| concepts[9].display_name | Coercivity |
| concepts[10].id | https://openalex.org/C118615104 |
| concepts[10].level | 1 |
| concepts[10].score | 0.38440629839897156 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q121416 |
| concepts[10].display_name | Discrete mathematics |
| concepts[11].id | https://openalex.org/C134306372 |
| concepts[11].level | 1 |
| concepts[11].score | 0.3061511516571045 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[11].display_name | Mathematical analysis |
| concepts[12].id | https://openalex.org/C121332964 |
| concepts[12].level | 0 |
| concepts[12].score | 0.18380168080329895 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[12].display_name | Physics |
| concepts[13].id | https://openalex.org/C41008148 |
| concepts[13].level | 0 |
| concepts[13].score | 0.09576916694641113 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[13].display_name | Computer science |
| concepts[14].id | https://openalex.org/C62520636 |
| concepts[14].level | 1 |
| concepts[14].score | 0.061768800020217896 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[14].display_name | Quantum mechanics |
| concepts[15].id | https://openalex.org/C34388435 |
| concepts[15].level | 2 |
| concepts[15].score | 0.04379221796989441 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q2267362 |
| concepts[15].display_name | Bounded function |
| concepts[16].id | https://openalex.org/C185592680 |
| concepts[16].level | 0 |
| concepts[16].score | 0.0 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q2329 |
| concepts[16].display_name | Chemistry |
| concepts[17].id | https://openalex.org/C162324750 |
| concepts[17].level | 0 |
| concepts[17].score | 0.0 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q8134 |
| concepts[17].display_name | Economics |
| concepts[18].id | https://openalex.org/C104317684 |
| concepts[18].level | 2 |
| concepts[18].score | 0.0 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q7187 |
| concepts[18].display_name | Gene |
| concepts[19].id | https://openalex.org/C86339819 |
| concepts[19].level | 3 |
| concepts[19].score | 0.0 |
| concepts[19].wikidata | https://www.wikidata.org/wiki/Q407384 |
| concepts[19].display_name | Transcription factor |
| concepts[20].id | https://openalex.org/C111919701 |
| concepts[20].level | 1 |
| concepts[20].score | 0.0 |
| concepts[20].wikidata | https://www.wikidata.org/wiki/Q9135 |
| concepts[20].display_name | Operating system |
| concepts[21].id | https://openalex.org/C41895202 |
| concepts[21].level | 1 |
| concepts[21].score | 0.0 |
| concepts[21].wikidata | https://www.wikidata.org/wiki/Q8162 |
| concepts[21].display_name | Linguistics |
| concepts[22].id | https://openalex.org/C158448853 |
| concepts[22].level | 4 |
| concepts[22].score | 0.0 |
| concepts[22].wikidata | https://www.wikidata.org/wiki/Q425218 |
| concepts[22].display_name | Repressor |
| concepts[23].id | https://openalex.org/C138885662 |
| concepts[23].level | 0 |
| concepts[23].score | 0.0 |
| concepts[23].wikidata | https://www.wikidata.org/wiki/Q5891 |
| concepts[23].display_name | Philosophy |
| concepts[24].id | https://openalex.org/C10138342 |
| concepts[24].level | 1 |
| concepts[24].score | 0.0 |
| concepts[24].wikidata | https://www.wikidata.org/wiki/Q43015 |
| concepts[24].display_name | Finance |
| concepts[25].id | https://openalex.org/C55493867 |
| concepts[25].level | 1 |
| concepts[25].score | 0.0 |
| concepts[25].wikidata | https://www.wikidata.org/wiki/Q7094 |
| concepts[25].display_name | Biochemistry |
| keywords[0].id | https://openalex.org/keywords/counterexample |
| keywords[0].score | 0.9429291486740112 |
| keywords[0].display_name | Counterexample |
| keywords[1].id | https://openalex.org/keywords/divergence |
| keywords[1].score | 0.8082431554794312 |
| keywords[1].display_name | Divergence (linguistics) |
| keywords[2].id | https://openalex.org/keywords/mathematics |
| keywords[2].score | 0.756725013256073 |
| keywords[2].display_name | Mathematics |
| keywords[3].id | https://openalex.org/keywords/space |
| keywords[3].score | 0.669793963432312 |
| keywords[3].display_name | Space (punctuation) |
| keywords[4].id | https://openalex.org/keywords/order |
| keywords[4].score | 0.6331356167793274 |
| keywords[4].display_name | Order (exchange) |
| keywords[5].id | https://openalex.org/keywords/operator |
| keywords[5].score | 0.5528900623321533 |
| keywords[5].display_name | Operator (biology) |
| keywords[6].id | https://openalex.org/keywords/pure-mathematics |
| keywords[6].score | 0.5158076286315918 |
| keywords[6].display_name | Pure mathematics |
| keywords[7].id | https://openalex.org/keywords/combinatorics |
| keywords[7].score | 0.4393959045410156 |
| keywords[7].display_name | Combinatorics |
| keywords[8].id | https://openalex.org/keywords/maximal-operator |
| keywords[8].score | 0.42741888761520386 |
| keywords[8].display_name | Maximal operator |
| keywords[9].id | https://openalex.org/keywords/coercivity |
| keywords[9].score | 0.42471766471862793 |
| keywords[9].display_name | Coercivity |
| keywords[10].id | https://openalex.org/keywords/discrete-mathematics |
| keywords[10].score | 0.38440629839897156 |
| keywords[10].display_name | Discrete mathematics |
| keywords[11].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[11].score | 0.3061511516571045 |
| keywords[11].display_name | Mathematical analysis |
| keywords[12].id | https://openalex.org/keywords/physics |
| keywords[12].score | 0.18380168080329895 |
| keywords[12].display_name | Physics |
| keywords[13].id | https://openalex.org/keywords/computer-science |
| keywords[13].score | 0.09576916694641113 |
| keywords[13].display_name | Computer science |
| keywords[14].id | https://openalex.org/keywords/quantum-mechanics |
| keywords[14].score | 0.061768800020217896 |
| keywords[14].display_name | Quantum mechanics |
| keywords[15].id | https://openalex.org/keywords/bounded-function |
| keywords[15].score | 0.04379221796989441 |
| keywords[15].display_name | Bounded function |
| language | en |
| locations[0].id | pmh:oai:arXiv.org:2401.05550 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400194 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | arXiv (Cornell University) |
| locations[0].source.host_organization | https://openalex.org/I205783295 |
| locations[0].source.host_organization_name | Cornell University |
| locations[0].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[0].license | |
| locations[0].pdf_url | https://arxiv.org/pdf/2401.05550 |
| locations[0].version | submittedVersion |
| locations[0].raw_type | text |
| locations[0].license_id | |
| locations[0].is_accepted | False |
| locations[0].is_published | False |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | http://arxiv.org/abs/2401.05550 |
| locations[1].id | doi:10.48550/arxiv.2401.05550 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | cc-by |
| locations[1].pdf_url | |
| locations[1].version | |
| locations[1].raw_type | article |
| locations[1].license_id | https://openalex.org/licenses/cc-by |
| locations[1].is_accepted | False |
| locations[1].is_published | |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://doi.org/10.48550/arxiv.2401.05550 |
| indexed_in | arxiv, datacite |
| authorships[0].author.id | https://openalex.org/A5004816364 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-1450-2285 |
| authorships[0].author.display_name | Sebastian Bechtel |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Bechtel, Sebastian |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5019324917 |
| authorships[1].author.orcid | https://orcid.org/0000-0003-1148-1973 |
| authorships[1].author.display_name | Connor Mooney |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Mooney, Connor |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5083145406 |
| authorships[2].author.orcid | https://orcid.org/0000-0003-3167-7471 |
| authorships[2].author.display_name | Mark Veraar |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Veraar, Mark |
| authorships[2].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://arxiv.org/pdf/2401.05550 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Counterexamples to maximal regularity for operators in divergence form |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T06:51:31.235846 |
| primary_topic.id | https://openalex.org/T11205 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.9980000257492065 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2610 |
| primary_topic.subfield.display_name | Mathematical Physics |
| primary_topic.display_name | Numerical methods in inverse problems |
| related_works | https://openalex.org/W2349952486, https://openalex.org/W1489260687, https://openalex.org/W1979864087, https://openalex.org/W2044642251, https://openalex.org/W3093027776, https://openalex.org/W1986796332, https://openalex.org/W2775700832, https://openalex.org/W2587222766, https://openalex.org/W3204670144, https://openalex.org/W2163101548 |
| cited_by_count | 0 |
| locations_count | 2 |
| best_oa_location.id | pmh:oai:arXiv.org:2401.05550 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400194 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | arXiv (Cornell University) |
| best_oa_location.source.host_organization | https://openalex.org/I205783295 |
| best_oa_location.source.host_organization_name | Cornell University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://arxiv.org/pdf/2401.05550 |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | text |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://arxiv.org/abs/2401.05550 |
| primary_location.id | pmh:oai:arXiv.org:2401.05550 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400194 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | arXiv (Cornell University) |
| primary_location.source.host_organization | https://openalex.org/I205783295 |
| primary_location.source.host_organization_name | Cornell University |
| primary_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| primary_location.license | |
| primary_location.pdf_url | https://arxiv.org/pdf/2401.05550 |
| primary_location.version | submittedVersion |
| primary_location.raw_type | text |
| primary_location.license_id | |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | http://arxiv.org/abs/2401.05550 |
| publication_date | 2024-01-10 |
| publication_year | 2024 |
| referenced_works_count | 0 |
| abstract_inverted_index.a | 10, 16, 42 |
| abstract_inverted_index.In | 0 |
| abstract_inverted_index.It | 27 |
| abstract_inverted_index.We | 57 |
| abstract_inverted_index.in | 19, 53, 60 |
| abstract_inverted_index.is | 15, 28 |
| abstract_inverted_index.on | 39, 45, 67, 71 |
| abstract_inverted_index.or | 69 |
| abstract_inverted_index.to | 6 |
| abstract_inverted_index.we | 3 |
| abstract_inverted_index.The | 13 |
| abstract_inverted_index.and | 24, 48, 55 |
| abstract_inverted_index.any | 50 |
| abstract_inverted_index.for | 9 |
| abstract_inverted_index.one | 62 |
| abstract_inverted_index.the | 46 |
| abstract_inverted_index.PDE. | 12 |
| abstract_inverted_index.form | 21 |
| abstract_inverted_index.from | 30 |
| abstract_inverted_index.show | 58 |
| abstract_inverted_index.such | 34 |
| abstract_inverted_index.that | 33, 59 |
| abstract_inverted_index.this | 1 |
| abstract_inverted_index.time | 54 |
| abstract_inverted_index.with | 22 |
| abstract_inverted_index.admit | 36 |
| abstract_inverted_index.space | 23 |
| abstract_inverted_index.under | 41 |
| abstract_inverted_index.Lions' | 31 |
| abstract_inverted_index.cannot | 63 |
| abstract_inverted_index.expect | 64 |
| abstract_inverted_index.paper, | 2 |
| abstract_inverted_index.space. | 56 |
| abstract_inverted_index.theory | 32 |
| abstract_inverted_index.example | 14 |
| abstract_inverted_index.general | 61 |
| abstract_inverted_index.maximal | 7, 37, 65 |
| abstract_inverted_index.present | 4 |
| abstract_inverted_index.without | 49 |
| abstract_inverted_index.$H^{-1}$ | 40 |
| abstract_inverted_index.operator | 18 |
| abstract_inverted_index.condition | 44 |
| abstract_inverted_index.operators | 35 |
| abstract_inverted_index.parabolic | 11 |
| abstract_inverted_index.coercivity | 43 |
| abstract_inverted_index.conditions | 52 |
| abstract_inverted_index.divergence | 20 |
| abstract_inverted_index.regularity | 51 |
| abstract_inverted_index.well-known | 29 |
| abstract_inverted_index.second-order | 17 |
| abstract_inverted_index.coefficients, | 47 |
| abstract_inverted_index.coefficients. | 26 |
| abstract_inverted_index.time-dependent | 25 |
| abstract_inverted_index.counterexamples | 5 |
| abstract_inverted_index.$L^2$-regularity | 38, 70 |
| abstract_inverted_index.$L^p$-regularity | 8, 66 |
| abstract_inverted_index.$L^2(\mathbb{R}^d)$. | 72 |
| abstract_inverted_index.$H^{-1}(\mathbb{R}^d)$ | 68 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile |