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[{"key": "dc.contributor.advisor", "value": "Kilpel\u00e4inen, Tero", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Siltakoski, Jarkko", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-05-05T17:36:26Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-05-05T17:36:26Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1532849", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/49651", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoitus on tutustua ep\u00e4lineaarisiin ominaisarvo-ongelmiin p-Laplacen operaattorin ominaisarvo-ongelman kautta. p-Laplacen operaattori on Laplacen operaattorin er\u00e4s yleistys ja tarkastelun kohteena oleva ominaisarvo-ongelma on Dirichletin ominaisarvo-ongelman yleistys.\nTutkielmassa kerrataan ensin tarvittavia taustatietoja Sobolevin avaruuksista ja funktionaalianalyysist\u00e4, ja keskityt\u00e4\u00e4n sitten itse ongelmaan. P\u00e4\u00e4tulokset koskevat ensimm\u00e4ist\u00e4 ominaisarvoa, ja ne ovat ensimm\u00e4isen ominaisarvon olemassaolo, ensimm\u00e4isen ominaisarvon karakterisointi Rayleighin osam\u00e4\u00e4r\u00e4n avulla, ensimm\u00e4isen ominaisfunktion yksinkertaisuus, ja se, ett\u00e4 ensimm\u00e4inen ominaisfunktio on ainoa ominaisfunktio, joka ei vaihda merkki\u00e4. Muita tuloksia ovat ominaisfunktioiden jatkuvuus ja rajoittuneisuus sek\u00e4 se, ett\u00e4 ominaisarvojen joukko on suljettu. Lis\u00e4ksi tutkielmassa k\u00e4yd\u00e4\u00e4n l\u00e4pi lineaarista erikoistapausta p=2, jossa p-Laplacen operaattori on tavallinen Laplacen operaattori.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Jarkko Siltakoski (jajamisi) on 2016-05-05 17:36:26.206532. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-05-05T17:36:26Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201605052437.pdf: 440988 bytes, checksum: 38690b209bc2753f1ec34d13cf368174 (MD5)\nlicense.html: 4809 bytes, checksum: 415a255a0842998fd1821836f0150d3f (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-05-05T17:36:26Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201605052437.pdf: 440988 bytes, checksum: 38690b209bc2753f1ec34d13cf368174 (MD5)\nlicense.html: 4809 bytes, checksum: 415a255a0842998fd1821836f0150d3f (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (51 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Laplace", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", 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