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[{"key": "dc.contributor.advisor", "value": "Parviainen, Mikko", "language": null, "element": "contributor", "schema": "dc", "qualifier": "advisor"}, {"key": "dc.contributor.author", "value": "Vilkki, Saana", "language": null, "element": "contributor", "schema": "dc", "qualifier": "author"}, {"key": "dc.date.accessioned", "value": "2025-03-14T06:19:59Z", "language": null, "element": "date", "schema": "dc", "qualifier": "accessioned"}, {"key": "dc.date.available", "value": "2025-03-14T06:19:59Z", "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.uri", "value": "https://jyx.jyu.fi/handle/123456789/100740", "language": null, "element": "identifier", "schema": "dc", "qualifier": "uri"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa osoitetaan p-Laplacen yht\u00e4l\u00f6n ratkaisujen olemassaolo ja\nyksik\u00e4sitteisyys. T\u00e4m\u00e4 ongelma on erikoistapaus Hilbertin 20. ongelmasta,\njoka k\u00e4sittelee variaatiolaskennan s\u00e4\u00e4nn\u00f6llisten ongelmien ratkaisujen olemassaoloa. Kyseinen yht\u00e4l\u00f6 on klassisen Laplacen yht\u00e4l\u00f6n \u2206u = 0 yleistys,\nja molemmilla yht\u00e4l\u00f6ill\u00e4 on laaja sovellettavuus niin matematiikassa kuin\nfysiikassa.\nTutkielman alussa muodostetaan perusta differentiaaliyht\u00e4l\u00f6iden tarkastelulle. Keskeisen\u00e4 tavoitteena on m\u00e4\u00e4ritell\u00e4 L\np\n-avaruudet ja Sobolevin avaruudet sek\u00e4 osoitetaan H\u00f6lderin ep\u00e4yht\u00e4l\u00f6, jota hy\u00f6dynnet\u00e4\u00e4n tutkielmassa\nuseissa kohdissa. Sobolevin avaruuksien avulla m\u00e4\u00e4ritell\u00e4\u00e4n heikot ratkaisut\nLaplacen ja p-Laplacen yht\u00e4l\u00f6ille. Sobolevin avaruuksia k\u00e4sitelless\u00e4 keskeisimpin\u00e4 tuloksina osoitetaan Sobolevin avaruuden t\u00e4ydellisyys ja m\u00e4\u00e4ritell\u00e4\u00e4n Sobolevin nolla-avaruus.\nTutkielman loppuosassa k\u00e4sitell\u00e4\u00e4n Laplacen yht\u00e4l\u00f6n ja p-Laplacen yht\u00e4l\u00f6n heikkoja ratkaisuja variaatiointegraalin avulla. Ratkaisujen olemassaolon\ntodistamista varten osoitetaan, ett\u00e4 yht\u00e4l\u00f6n variaatiointegraalin minimoijat\novat samat kuin heikot ratkaisut.", "language": "fi", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "schema": "dc", "qualifier": "abstract"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-03-14T06:19:59Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-03-14T06:19:59Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "schema": "dc", "qualifier": "provenance"}, {"key": "dc.format.extent", "value": "33", "language": null, "element": "format", "schema": "dc", "qualifier": "extent"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "schema": "dc", "qualifier": "iso"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "schema": "dc", "qualifier": null}, {"key": "dc.subject.other", "value": "", "language": null, "element": "subject", "schema": "dc", "qualifier": "other"}, {"key": "dc.title", "value": "Minimoijat ja p-Laplacen yht\u00e4l\u00f6", "language": null, "element": "title", "schema": "dc", "qualifier": null}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "schema": "dc", "qualifier": null}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202503142553", "language": null, "element": "identifier", "schema": "dc", "qualifier": "urn"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "schema": "dc", "qualifier": "faculty"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "schema": "dc", "qualifier": "department"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "schema": "dc", "qualifier": "department"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "schema": "dc", "qualifier": "organization"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "schema": "dc", "qualifier": "organization"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "schema": "dc", "qualifier": "discipline"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "schema": "dc", "qualifier": "discipline"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "schema": "dc", "qualifier": "coar"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "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": "masterThesis", "language": null, "element": "type", "schema": "dc", "qualifier": "publication"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "osittaisdifferentiaaliyht\u00e4l\u00f6t", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.subject.yso", "value": "variaatiolaskenta", "language": null, "element": "subject", "schema": "dc", "qualifier": "yso"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "schema": "dc", "qualifier": "url"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "schema": "dc", "qualifier": "accessibilityfeature"}]
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