{"id":308,"date":"2025-07-08T13:55:53","date_gmt":"2025-07-08T12:55:53","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=308"},"modified":"2025-07-08T14:01:24","modified_gmt":"2025-07-08T13:01:24","slug":"teoremes-dextensio","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/teoremes-dextensio\/","title":{"rendered":"Teoremes d\u2019extensi\u00f3"},"content":{"rendered":"\n<p><strong>Prerrequisit<\/strong>: pels continguts tractats en el TFG, \u00e9s important haver estudiat teoria de la mesura (integral de Lebesgue com a m\u00ednim).<\/p>\n<p>Estudiarem operadors d\u2019extensi\u00f3, \u00e9s a dir, formes d\u2019estendre una funci\u00f3 definida en un cert domini a l\u2019espai ambient (amb derivades integrables localment), de manera que la norma de la funci\u00f3 resultant estigui controlada per la norma inicial. Comen\u00e7arem per l&#8217;espai de funcions \\(W^{k,p}\\) (funcions amb \\(k\\) derivades febles \\(p\\)-integrables), per\u00f2 si es creu convenient ens podem centrar en espais de suavitat fraccion\u00e0ria, per exemple.<\/p>\n<p>Els continguts proposats s\u00f3n:<\/p>\n<ul>\n<li>Espais de Sobolev [1]<\/li>\n<li>Extensions per a dominis regulars [2]<\/li>\n<li>Extensions per dominis menys regulars (Lipschitz i uniformes) [3] i altres de m\u00e9s avan\u00e7ats depenent de l&#8217;evoluci\u00f3.<\/li>\n<li>Espais de Triebel-Lizorkin o de Besov si es vol tirar cap aqu\u00ed, de nivell m\u00e9s avan\u00e7at.<\/li>\n<\/ul>\n<p>[1] Evans, Lawrence C. <em>Partial di\ufb00erential equations<\/em>. Graduate Studies in Mathematics 19 (1998).<\/p>\n<p>[2] Stein, Elias M. <em>Singular integrals and differentiability properties of functions (PMS-30)<\/em>. Vol. 30. Princeton university press, 1970<\/p>\n<p>[3] Jones, Peter W. &#8220;Quasiconformal mappings and extendability of functions in Sobolev spaces.&#8221; <em>Acta Mathematica<\/em> 147.1 (1981): 71-88.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Prerrequisit: pels continguts tractats en el TFG, \u00e9s important haver estudiat teoria de la mesura (integral de Lebesgue com a m\u00ednim). Estudiarem operadors d\u2019extensi\u00f3, \u00e9s a dir, formes d\u2019estendre una funci\u00f3 definida en un cert domini a l\u2019espai ambient (amb derivades integrables localment), de manera que la norma de la funci\u00f3 resultant estigui controlada per [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[40],"class_list":["post-308","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-marti-prats"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/308","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=308"}],"version-history":[{"count":6,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/308\/revisions"}],"predecessor-version":[{"id":314,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/308\/revisions\/314"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=308"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=308"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=308"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}