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[{"key": "dc.contributor.author", "value": "Yawen, Feng", "language": null, "element": "contributor", "schema": "dc", "qualifier": "author"}, {"key": "dc.date.accessioned", "value": "2025-05-27T11:11:05Z", "language": null, "element": "date", "schema": "dc", "qualifier": "accessioned"}, {"key": "dc.date.available", "value": "2025-05-27T11:11:05Z", "language": null, "element": "date", "schema": "dc", "qualifier": "available"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "schema": "dc", "qualifier": "issued"}, {"key": "dc.identifier.isbn", "value": "978-952-86-0752-6", "language": null, "element": "identifier", "schema": "dc", "qualifier": "isbn"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/102795", "language": null, "element": "identifier", "schema": "dc", "qualifier": "uri"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 v\u00e4it\u00f6skirjassa tutkitaan s\u00e4\u00e4nn\u00f6llisyysteoriaa parabolisille osittaisdifferentiaaliyht\u00e4l\u00f6ille. Erityisesti tarkastellaan viskositeettiratkaisujen s\u00e4\u00e4nn\u00f6llisyysominaisuuksia yleistetylle p-paraboliselle yht\u00e4l\u00f6lle. Sek\u00e4 tavallinen p-parabolinen yht\u00e4l\u00f6 ett\u00e4 normalisoitu p-parabolinen yht\u00e4l\u00f6 ovat yleistetyn yht\u00e4l\u00f6n erikoistapauksia. Tarkemmin sanottuna ty\u00f6ss\u00e4 tutkitaan yleistetyn p-parabolisen yht\u00e4l\u00f6n ratkaisujentoisen kertaluvun s\u00e4\u00e4nn\u00f6llisyytt\u00e4. V\u00e4it\u00f6skirja koostuu johdannosta ja kolmesta tutkimusartikkelista. Artikkelit on julkaistu lehdiss\u00e4 Journal of Evolution Equations, Calculus of Variations and Partial Differential Equations sek\u00e4 Journal of Functional Analysis. Ensimm\u00e4isess\u00e4 artikkelissa tutkitaan p-parabolisten funktioiden, jotka ovat p-parabolisen yht\u00e4l\u00f6n heikkoja ratkaisuja, lokaalia toisen kertaluvun Sobolevin s\u00e4\u00e4nn\u00f6llisyytt\u00e4. Vastaesimerkki osoittaa, ett\u00e4 artikkelissa k\u00e4ytetty parametrien alue on tarkka. Toisessa artikkelissa laajennetaan tarkastelua p-parabolisesta yht\u00e4l\u00f6st\u00e4 yleisemp\u00e4\u00e4n tapaukseen. Erityisesti artikkelissa todistetaan yleistetyn p-parabolisen yht\u00e4l\u00f6n viskositeettiratkaisujen lokaalitoisen kertaluvun s\u00e4\u00e4nn\u00f6llisyys tasossa tietyille sallituille parametreille. Lis\u00e4ksi ty\u00f6ss\u00e4 kehitet\u00e4\u00e4n systemaattinen menetelm\u00e4, jossa kysymys toisen kertaluvun s\u00e4\u00e4nn\u00f6llisyydest\u00e4 palautuu tietyn kerroinmatriisin positiivisuuden todistamiseen. Viimeisess\u00e4 artikkelissa parannetaan parametrien alueita sek\u00e4 tutkitaan er\u00e4\u00e4n ep\u00e4lineaarisen suureen lokaalia s\u00e4\u00e4nn\u00f6llisyytt\u00e4 tasossa yleistetylle p-paraboliselle yht\u00e4l\u00f6lle. T\u00e4m\u00e4 ep\u00e4lineaarinen suure on sama kuin ensimm\u00e4isess\u00e4 artikkelissa p-parabolisille funktioille tarkasteltu suure. Todistusten keskeinen idea perustuu divergenssirakenteiden l\u00f6yt\u00e4miseen ja hy\u00f6dynt\u00e4miseen sek\u00e4 er\u00e4\u00e4seen ep\u00e4yht\u00e4l\u00f6\u00f6n. Tietyt toisen kertaluvun termit, jotka aluksi eiv\u00e4t n\u00e4yt\u00e4 olevan divergenssimuodossa, voidaan itse asiassa kirjoittaa divergenssimuotoon.", "language": "fin", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.abstract", "value": "This dissertation studies regularity theory for solutions to parabolic partial differential equations. We investigate regularity properties of viscosity solutions to the generalized p-parabolicequation. Both the standard p-parabolic equation and the normalized p-parabolic equation are special cases of this equation. We analyze the second-order regularity of solutions to the generalized p-parabolic equation. This work comprises a general introduction and three research papers. The papers are published in Journal of Evolution Equations, Calculus of Variations and Partial Differential Equations, and Journal of Functional Analysis. In the first article, we study the local second-order Sobolev regularity of p-parabolic functions, which are weak solutions to the p-parabolic equation. A counterexample shows that the obtained range is sharp. In the second article, we extend the study from the p-parabolic equation to the more general case. We prove the local second-order regularity for planar viscosity solutions to the generalized p-parabolic equation in certain ranges. We also establish a systematic approach where the question about the second-order regularity boils down to verifying that a certain coefficient matrix is positive definite. In the last article, we improve these ranges. We also prove the local regularity of the nonlinear quantity for the generalized p-parabolic equation in the plane. This nonlinear quantity is the same as the one considered for p-parabolic functions in the first article. A key idea involves the use of hidden divergence structures and a fundamental inequality. Some second-order terms that first appear in non-divergence form can actually be rewritten in divergence form.", "language": "en", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.provenance", "value": "Submitted by Timo Hautala (tikaha@jyu.fi) on 2025-05-27T11:11:05Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-05-27T11:11:05Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "schema": "dc", "qualifier": "mimetype"}, {"key": "dc.publisher", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "publisher", "schema": "dc", "qualifier": null}, {"key": "dc.relation.ispartofseries", "value": "JYU Dissertations", "language": null, "element": "relation", "schema": "dc", "qualifier": "ispartofseries"}, {"key": "dc.title", "value": "Second-order Sobolev regularity for a class of generalized nonlinear p-parabolic equations", "language": null, "element": "title", "schema": "dc", "qualifier": null}, {"key": "dc.type", "value": "doctoral thesis", "language": null, "element": "type", "schema": "dc", "qualifier": null}, {"key": "dc.identifier.urn", "value": "URN:ISBN:978-952-86-0752-6", "language": null, "element": "identifier", "schema": "dc", "qualifier": "urn"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": null, "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.contributor.faculty", "value": "Faculty of Mathematics and Science", "language": null, "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_db06", "language": null, "element": "type", "schema": "dc", "qualifier": "coar"}, {"key": "dc.relation.issn", "value": "2489-9003", "language": null, "element": "relation", "schema": "dc", "qualifier": "issn"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author & University of Jyv\u00e4skyl\u00e4", "language": null, "element": "rights", "schema": "dc", "qualifier": "copyright"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "schema": "dc", "qualifier": "accesslevel"}, {"key": "dc.type.publication", "value": "doctoralThesis", "language": null, "element": "type", "schema": "dc", "qualifier": "publication"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "schema": "dc", "qualifier": "content"}, {"key": "dc.description.accessibilityfeature", "value": "kuvilla vaihtoehtoiset kuvaukset", "language": "fi", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}, {"key": "dc.description.accessibilityfeature", "value": "alternative textual descriptions for images", "language": "en", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}, {"key": "dc.description.accessibilityfeature", "value": "navigointi mahdollista", "language": "fi", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}, {"key": "dc.description.accessibilityfeature", "value": "structural navigation", "language": "en", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}, {"key": "dc.description.accessibilitysummary", "value": "PDF tehty LaTeX-ohjelmalla, dokumentti ei v\u00e4ltt\u00e4m\u00e4tt\u00e4 ole t\u00e4ysin saavutettava.", "language": "fi", "element": "description", "schema": "dc", "qualifier": "accessibilitysummary"}, {"key": "dc.description.accessibilitysummary", "value": "Made with LaTeX, not fully accessible document.", "language": "en", "element": "description", "schema": "dc", "qualifier": "accessibilitysummary"}]
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