{"id":"https://openalex.org/W4221156457","doi":"https://doi.org/10.1007/s00332-022-09820-x","title":"Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey\u2019s Conjecture","display_name":"Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey\u2019s Conjecture","publication_year":2022,"publication_date":"2022-09-02","ids":{"openalex":"https://openalex.org/W4221156457","doi":"https://doi.org/10.1007/s00332-022-09820-x"},"language":"en","primary_location":{"id":"doi:10.1007/s00332-022-09820-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09820-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09820-x.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09820-x.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043356572","display_name":"Jendrik Voss","orcid":"https://orcid.org/0000-0001-8912-9625"},"institutions":[{"id":"https://openalex.org/I200332995","display_name":"TU Dortmund University","ror":"https://ror.org/01k97gp34","country_code":"DE","type":"education","lineage":["https://openalex.org/I200332995"]},{"id":"https://openalex.org/I62318514","display_name":"University of Duisburg-Essen","ror":"https://ror.org/04mz5ra38","country_code":"DE","type":"education","lineage":["https://openalex.org/I62318514"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Jendrik Voss","raw_affiliation_strings":["Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany","Institute for Structural Mechanics and Dynamics, Technical University Dortmund, August-Schmidt-Str. 8, 44227, Dortmund, Germany"],"raw_orcid":"https://orcid.org/0000-0001-8912-9625","affiliations":[{"raw_affiliation_string":"Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany","institution_ids":["https://openalex.org/I62318514"]},{"raw_affiliation_string":"Institute for Structural Mechanics and Dynamics, Technical University Dortmund, August-Schmidt-Str. 8, 44227, Dortmund, Germany","institution_ids":["https://openalex.org/I200332995"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018069050","display_name":"R.J. Martin","orcid":"https://orcid.org/0000-0002-1155-6704"},"institutions":[{"id":"https://openalex.org/I62318514","display_name":"University of Duisburg-Essen","ror":"https://ror.org/04mz5ra38","country_code":"DE","type":"education","lineage":["https://openalex.org/I62318514"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Robert J. Martin","raw_affiliation_strings":["Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany","Institute for Technologies of Metals, University of Duisburg-Essen, Friedrich-Ebert-Str. 12, 47119, Duisburg, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany","institution_ids":["https://openalex.org/I62318514"]},{"raw_affiliation_string":"Institute for Technologies of Metals, University of Duisburg-Essen, Friedrich-Ebert-Str. 12, 47119, Duisburg, Germany","institution_ids":["https://openalex.org/I62318514"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Oliver Sander","orcid":"https://orcid.org/0000-0003-1093-6374"},"institutions":[{"id":"https://openalex.org/I78650965","display_name":"Technische Universit\u00e4t Dresden","ror":"https://ror.org/042aqky30","country_code":"DE","type":"education","lineage":["https://openalex.org/I78650965"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Oliver Sander","raw_affiliation_strings":["Institute of Numerical Mathematics, Technical University of Dresden, Zellescher Weg 12-14, 01069, Dresden, Germany"],"raw_orcid":"https://orcid.org/0000-0003-1093-6374","affiliations":[{"raw_affiliation_string":"Institute of Numerical Mathematics, Technical University of Dresden, Zellescher Weg 12-14, 01069, Dresden, Germany","institution_ids":["https://openalex.org/I78650965"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034720252","display_name":"Siddhant Kumar","orcid":"https://orcid.org/0000-0003-1602-8641"},"institutions":[{"id":"https://openalex.org/I98358874","display_name":"Delft University of Technology","ror":"https://ror.org/02e2c7k09","country_code":"NL","type":"education","lineage":["https://openalex.org/I98358874"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Siddhant Kumar","raw_affiliation_strings":["Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628CD, Delft, Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628CD, Delft, Netherlands","institution_ids":["https://openalex.org/I98358874"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083881460","display_name":"Dennis M. Kochmann","orcid":"https://orcid.org/0000-0002-9112-6615"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Dennis M. Kochmann","raw_affiliation_strings":["Mechanics and Materials Lab, Department of Mechanical and Process Engineering, ETH Z\u00fcrich, Leonhardstr. 21, 8092, Z\u00fcrich, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mechanics and Materials Lab, Department of Mechanical and Process Engineering, ETH Z\u00fcrich, Leonhardstr. 21, 8092, Z\u00fcrich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046930021","display_name":"Patrizio Neff","orcid":"https://orcid.org/0000-0002-1615-8879"},"institutions":[{"id":"https://openalex.org/I62318514","display_name":"University of Duisburg-Essen","ror":"https://ror.org/04mz5ra38","country_code":"DE","type":"education","lineage":["https://openalex.org/I62318514"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Patrizio Neff","raw_affiliation_strings":["Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, Germany","institution_ids":["https://openalex.org/I62318514"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5043356572"],"corresponding_institution_ids":["https://openalex.org/I200332995","https://openalex.org/I62318514"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.05290401,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"32","issue":"6","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.7075999975204468,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.7075999975204468,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11822","display_name":"Analytic and geometric function theory","score":0.19789999723434448,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.02019999921321869,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.6577507853507996},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6360827088356018},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.5735848546028137},{"id":"https://openalex.org/keywords/quasiconvex-function","display_name":"Quasiconvex function","score":0.560204267501831},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5565770268440247},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5240585803985596},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4939742684364319},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4797978699207306},{"id":"https://openalex.org/keywords/counterexample","display_name":"Counterexample","score":0.47883084416389465},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.4411989450454712},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.4348660111427307},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3576884865760803},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2803528904914856},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.11985728144645691},{"id":"https://openalex.org/keywords/subderivative","display_name":"Subderivative","score":0.10294294357299805},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0522305965423584}],"concepts":[{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.6577507853507996},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6360827088356018},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.5735848546028137},{"id":"https://openalex.org/C80278974","wikidata":"https://www.wikidata.org/wiki/Q1126412","display_name":"Quasiconvex function","level":5,"score":0.560204267501831},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5565770268440247},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5240585803985596},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4939742684364319},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4797978699207306},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.47883084416389465},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.4411989450454712},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4348660111427307},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3576884865760803},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2803528904914856},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.11985728144645691},{"id":"https://openalex.org/C200661725","wikidata":"https://www.wikidata.org/wiki/Q289500","display_name":"Subderivative","level":4,"score":0.10294294357299805},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0522305965423584},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s00332-022-09820-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09820-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09820-x.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2201.06392","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2201.06392","pdf_url":"https://arxiv.org/pdf/2201.06392","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:tudelft.nl:uuid:8007394a-c559-4e2c-b8ae-6338e7260ff4","is_oa":false,"landing_page_url":"http://resolver.tudelft.nl/uuid:8007394a-c559-4e2c-b8ae-6338e7260ff4","pdf_url":null,"source":{"id":"https://openalex.org/S4306400906","display_name":"Research Repository (Delft University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I98358874","host_organization_name":"Delft University of Technology","host_organization_lineage":["https://openalex.org/I98358874"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"journal article"},{"id":"pmh:oai:www.research-collection.ethz.ch:20.500.11850/571161","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11850/571161","pdf_url":null,"source":{"id":"https://openalex.org/S4306402302","display_name":"Repository for Publications and Research Data (ETH Zurich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I35440088","host_organization_name":"ETH Zurich","host_organization_lineage":["https://openalex.org/I35440088"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Nonlinear Science, 32 (6)","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"doi:10.48550/arxiv.2201.06392","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2201.06392","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.3929/ethz-b-000571161","is_oa":true,"landing_page_url":"https://doi.org/10.3929/ethz-b-000571161","pdf_url":null,"source":{"id":"https://openalex.org/S7407051236","display_name":"ETH Z\u00fcrich Research Collection","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s00332-022-09820-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09820-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09820-x.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320325052","display_name":"Universit\u00e4t Duisburg-Essen","ror":"https://ror.org/04mz5ra38"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4221156457.pdf","grobid_xml":"https://content.openalex.org/works/W4221156457.grobid-xml"},"referenced_works_count":59,"referenced_works":["https://openalex.org/W62770102","https://openalex.org/W147998453","https://openalex.org/W153666221","https://openalex.org/W175747256","https://openalex.org/W576274240","https://openalex.org/W1535449218","https://openalex.org/W1555984062","https://openalex.org/W1588906972","https://openalex.org/W1656819531","https://openalex.org/W1973385242","https://openalex.org/W1976872692","https://openalex.org/W1978018044","https://openalex.org/W1979167448","https://openalex.org/W1990237558","https://openalex.org/W1994484217","https://openalex.org/W2001215685","https://openalex.org/W2014212235","https://openalex.org/W2028074989","https://openalex.org/W2029134200","https://openalex.org/W2030579135","https://openalex.org/W2045970486","https://openalex.org/W2046930772","https://openalex.org/W2054695646","https://openalex.org/W2065659948","https://openalex.org/W2067244108","https://openalex.org/W2069241221","https://openalex.org/W2076063813","https://openalex.org/W2077140284","https://openalex.org/W2084478435","https://openalex.org/W2088703872","https://openalex.org/W2114821056","https://openalex.org/W2123322754","https://openalex.org/W2132995814","https://openalex.org/W2135837098","https://openalex.org/W2152394501","https://openalex.org/W2161397135","https://openalex.org/W2168396505","https://openalex.org/W2266240621","https://openalex.org/W2496722619","https://openalex.org/W2752438380","https://openalex.org/W2753441252","https://openalex.org/W2884658347","https://openalex.org/W2885361378","https://openalex.org/W2899283552","https://openalex.org/W2905184499","https://openalex.org/W2970971581","https://openalex.org/W2992001690","https://openalex.org/W3043048973","https://openalex.org/W3111771697","https://openalex.org/W3112401890","https://openalex.org/W3127287360","https://openalex.org/W3163993681","https://openalex.org/W3177248489","https://openalex.org/W3215085381","https://openalex.org/W4205634890","https://openalex.org/W4205876119","https://openalex.org/W4210831390","https://openalex.org/W4281252998","https://openalex.org/W4293194885"],"related_works":["https://openalex.org/W31957904","https://openalex.org/W15967914","https://openalex.org/W8335114","https://openalex.org/W21322171","https://openalex.org/W26139782","https://openalex.org/W25928676","https://openalex.org/W23243106","https://openalex.org/W21818054","https://openalex.org/W22671108","https://openalex.org/W6305162"],"abstract_inverted_index":{"Abstract":[0],"Deciding":[1],"whether":[2,320],"a":[3,10,16,29,35,167,303],"given":[4,36],"function":[5,37,51,177],"is":[6,8,309],"quasiconvex":[7],"generally":[9],"difficult":[11],"task.":[12],"Here,":[13],"we":[14,170],"discuss":[15],"number":[17],"of":[18,34,159,175,217,311],"numerical":[19],"approaches":[20],"that":[21,173],"can":[22],"be":[23],"used":[24],"in":[25,327],"the":[26,32,46,156,218,328],"search":[27],"for":[28,181],"counterexample":[30],"to":[31,316],"quasiconvexity":[33,174,180,326],"W":[38],".":[39],"We":[40],"will":[41],"demonstrate":[42],"these":[43],"methods":[44],"using":[45],"planar":[47,186,329],"isotropic":[48,185],"rank-one":[49,183,323],"convex":[50,184],"$$\\begin{aligned}":[52,220],"W_\\textrm{magic}^+(F)=\\frac{\\lambda":[53],"_\\textrm{max}}{\\lambda":[54,57,61],"_\\textrm{min}}-\\log":[55],"\\frac{\\lambda":[56],"_\\textrm{min}}+\\log":[58],"\\det":[59],"F=\\frac{\\lambda":[60],"_\\textrm{min}}+2\\log":[62],"\\lambda":[63,139],"_\\textrm{min}\\,,":[64],"\\end{aligned}$$":[65,227],"<mml:math":[66,141,192,228],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[67,142,193,229],"<mml:mrow>":[68,72,78,143,194,198,203,230,234,236,246,256,270,276,282,290],"<mml:mtable>":[69,231],"<mml:mtr>":[70,232],"<mml:mtd>":[71,233],"<mml:msubsup>":[73],"<mml:mi>W</mml:mi>":[74,195,235,243,253,265,287],"<mml:mtext>magic</mml:mtext>":[75],"<mml:mo>+</mml:mo>":[76,106,121,201,251,285],"</mml:msubsup>":[77],"<mml:mo>(</mml:mo>":[79,204,237,247,257,271,291],"<mml:mi>F</mml:mi>":[80,109,238,248,259,274,278,293],"<mml:mo>)</mml:mo>":[81,206,239,249,260,283,294],"</mml:mrow>":[82,130,134,153,200,207,210,240,250,261,272,279,284,295,296,300],"<mml:mo>=</mml:mo>":[83,110,241,262],"<mml:mfrac>":[84,96,111,273],"<mml:msub>":[85,89,97,101,112,116,124,144,149,242,252,263,286],"<mml:mi>\u03bb</mml:mi>":[86,90,98,102,113,117,125,145,150],"<mml:mtext>max</mml:mtext>":[87,99,114,146],"</mml:msub>":[88,92,100,104,115,119,127,147,152,245,255,269,289],"<mml:mtext>min</mml:mtext>":[91,103,118,126,151],"</mml:mfrac>":[93,105,120,281],"<mml:mo>-</mml:mo>":[94],"<mml:mo>log</mml:mo>":[95,107,123],"<mml:mo>det</mml:mo>":[108,258,277,292],"<mml:mn>2</mml:mn>":[122,205],"<mml:mspace/>":[128],"<mml:mo>,</mml:mo>":[129],"</mml:mtd>":[131,297],"</mml:mtr>":[132,298],"</mml:mtable>":[133,299],"</mml:math>":[135,154,211,301],"where":[136],"$$\\lambda":[137],"_\\textrm{max}\\ge":[138],"_\\textrm{min}$$":[140],"<mml:mo>\u2265</mml:mo>":[148],"are":[155],"singular":[157],"values":[158],"F":[160],",":[161],"as":[162],"our":[163],"main":[164],"example.":[165],"In":[166],"previous":[168],"contribution,":[169],"have":[171],"shown":[172],"this":[176],"would":[178],"imply":[179],"all":[182],"energies":[187],"$$W:{\\text":[188],"{GL}}^+(2)\\rightarrow":[189],"{\\mathbb":[190],"{R}}$$":[191],"<mml:mo>:</mml:mo>":[196],"<mml:msup>":[197],"<mml:mtext>GL</mml:mtext>":[199],"</mml:msup>":[202],"<mml:mo>\u2192</mml:mo>":[208],"<mml:mi>R</mml:mi>":[209],"with":[212,302,314],"an":[213],"additive":[214],"volumetric-isochoric":[215],"split":[216],"form":[219],"W(F)=W_\\textrm{iso}(F)+W_\\textrm{vol}(\\det":[221],"F)={\\widetilde{W}}_\\textrm{iso}\\bigg":[222],"(\\frac{F}{\\sqrt{\\det":[223],"F}}\\bigg":[224],")+W_\\textrm{vol}(\\det":[225],"F)":[226],"<mml:mtext>iso</mml:mtext>":[244,268],"<mml:mtext>vol</mml:mtext>":[254,288],"<mml:mover>":[264],"<mml:mo>~</mml:mo>":[266],"</mml:mover>":[267],"<mml:msqrt>":[275],"</mml:msqrt>":[280],"concave":[304],"volumetric":[305],"part.":[306],"This":[307],"example":[308],"therefore":[310],"particular":[312],"interest":[313],"regard":[315],"Morrey\u2019s":[317],"open":[318],"question":[319],"or":[321],"not":[322],"convexity":[324],"implies":[325],"case.":[330]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2022-04-03T00:00:00"}