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[{"key": "dc.contributor.advisor", "value": "Julin, Vesa", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Yli-Sorvari, Arttu", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-08-27T07:35:42Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-08-27T07:35:42Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/59336", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Todistetaan variaatio-ongelmissa hy\u00f6dyllinen ja hyvin tunnettu\r\nP\u00f3lyan ja Szeg\u0151n ep\u00e4yht\u00e4l\u00f6, jonka mukaan Dirichlet'n\r\n\\(p\\)-energia pienenee Sobolev-funktioiden Schwarzin symmetrisoinnissa.\r\nT\u00e4m\u00e4 voidaan muotoilla siten, ett\u00e4 jos \\(u \\in W^{1,p}(\\mathbb{R}^n)\\),\r\n\\(1 \\le p < \\infty\\) ja \\(u^\\star\\) on funktion \\(u\\) v\u00e4henev\u00e4 pallosymmetrinen\r\nuudelleenj\u00e4rjestys, niin p\u00e4tee\r\n\\[\r\n\\int_{\\mathbb{R}^n} |\\nabla u^\\star|^p\\,dm_n \\le \\int_{\\mathbb{R}^n} |\\nabla u|^p\\,dm_n.\r\n\\]\r\n\r\nSchwarzin symmetrisoinnissa funktiota \\(u\\) kohti muodostetaan siis pallosymmetrinen\r\nja v\u00e4henev\u00e4 funktio \\(u^\\star\\) muuttamatta sen distribuutiofunktiota.\r\nN\u00e4ist\u00e4 ominaisuuksista seuraa, ett\u00e4 alkukuvajoukot \\(\\{u^\\star > t\\}\\) ovat joukkojen\r\n\\(\\{u > t\\}\\) kanssa samanmittaisia palloja. T\u00e4m\u00e4 taas liitt\u00e4\u00e4 symmetrisoinnin\r\nluontevasti isoperimetriseen ep\u00e4yht\u00e4l\u00f6\u00f6n, jonka mukaan avaruuden \\(\\mathbb{R}^n\\)\r\nsamanmittaisista (ja \u00e4\u00e4rellismittaisista) osajoukoista palloilla on pienin reunan pintamitta.\r\nT\u00e4m\u00e4 on my\u00f6s perimm\u00e4inen syy sille, miksi P\u00f3lyan ja Szeg\u0151n ep\u00e4yht\u00e4l\u00f6 on tosi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Riitta Pitk\u00e4nen (rpitkane@jyu.fi) on 2018-08-27T07:35:42Z\r\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-08-27T07:35:42Z (GMT). No. of bitstreams: 0\r\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "69", "language": "", "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": "uudelleenj\u00e4rjestys", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Schwarzin symmetrisointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "P\u00f3lyan ja Szeg\u0151n ep\u00e4yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "isoperimetrinen ep\u00e4yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Symmetrisointi ja P\u00f3lya-Szeg\u0151-ep\u00e4yht\u00e4l\u00f6", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201808273938", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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