{"id":197,"date":"2025-07-02T11:06:07","date_gmt":"2025-07-02T10:06:07","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=197"},"modified":"2025-07-02T11:06:07","modified_gmt":"2025-07-02T10:06:07","slug":"espais-de-sobolev-i-desigualtats-de-sobolev","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/espais-de-sobolev-i-desigualtats-de-sobolev\/","title":{"rendered":"Espais de Sobolev i desigualtats de Sobolev"},"content":{"rendered":"

Un espai de Sobolev<\/strong> \u00e9s un espai funcional que generalitza el concepte de derivada a funcions que no s\u00f3n necess\u00e0riament diferenciables en el sentit cl\u00e0ssic, per\u00f2 s\u00ed en un sentit feble<\/strong> o distribucional<\/strong>.<\/p>\n

Formalment, per a un conjunt obert \\( \\Omega \\subseteq \\mathbb{R}^n \\), \\( k \\in \\mathbb{N} \\) i \\( 1 \\leq p \\leq \\infty \\). L’espai de Sobolev \\( W^{k,p}(\\Omega) \\) es defineix com el conjunt de funcions \\( u \\in L^p(\\Omega) \\) tals que totes les seves derivades febles fins a ordre \\( k \\) tamb\u00e9 pertanyen a \\(L^p(\\Omega) \\). \u00c9s a dir:<\/p>\n

$$
\nW^{k,p}(\\Omega) = \\left\\{ u \\in L^p(\\Omega) \\;\\middle|\\; D^\\alpha u \\in L^p(\\Omega),\\; \\forall \\alpha \\in \\mathbb{N}^n \\text{ amb }
\n|\\alpha| \\leq k \\right\\}
\n$$<\/p>\n

on \\( D^\\alpha u \\) denota la derivada feble d\u2019ordre multi\u00edndex \\( \\alpha \\).<\/p>\n

Quan p = 2 , l’espai \\( W^{k,2}(\\Omega) \\) \u00e9s un espai de Hilbert, i sovint s’anomena \\( H^k(\\Omega) \\).<\/p>\n

Els espais de Sobolev s\u00f3n fonamentals en\u00a0 l\u2019an\u00e0lisi funcional, les equacions en derivades parcials (EDP),\u00a0 el c\u00e0lcul de variacions,\u00a0 la teoria de l\u2019aproximaci\u00f3, etc.<\/p>\n

Estudiarem les propietats funcionals d’aquests espais, el concepte de
\nderivada feble i les desigualtats de Sobolev que s\u00f3n una eina fonamental en l’an\u00e0lisi i en l’estudi de les Equacions en Derivades Parcials (EDP). Aquestes desigualtats expressen el fet sorprenent que \u00e9s possible controlar la mida d’una funci\u00f3 si controlem la mida de les seves derivades. M\u00e9s precisament tenim resultats del tipus:<\/p>\n

Sigui \\(0<k<n\/p\\) i \\(1\\leq p<\\infty \\). Si \\(1\/p-1\/q=k\/n\\), llavors l’espai de
\nSobolev \\(W^{k,p}(\\mathbb{R}^{n})\\) est\u00e0 cont\u00ednuament contingut a l’espai de Lebesgue \\(L^{q}\\left( \\mathbb{R}^{n}\\right) \\) \\'{e}s a dir
\n$$
\n\\left\\| f\\right\\| _{L^{q}\\left( \\mathbb{R}^{n}\\right) }\\leq C\\left\\|
\nf\\right\\| _{W^{k,p}(\\mathbb{R}^{n})}, f\\in W^{k,p}(\\mathbb{R}^{n})
\n$$
\non \\(\\left\\| f\\right\\| _{W^{k,p}(\\mathbb{R}^{n})}=\\sum_{\\left| \\alpha \\right|
\n}\\left\\| D^{\\alpha }f\\right\\| _{L^{p}\\left( \\mathbb{R}^{n}\\right) }.\\)<\/p>\n","protected":false},"excerpt":{"rendered":"

Un espai de Sobolev \u00e9s un espai funcional que generalitza el concepte de derivada a funcions que no s\u00f3n necess\u00e0riament diferenciables en el sentit cl\u00e0ssic, per\u00f2 s\u00ed en un sentit feble o distribucional. Formalment, per a un conjunt obert \\( \\Omega \\subseteq \\mathbb{R}^n \\), \\( k \\in \\mathbb{N} \\) i \\( 1 \\leq p \\leq […]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[37],"class_list":["post-197","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-joaquim-martin"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/197","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\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=197"}],"version-history":[{"count":6,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/197\/revisions"}],"predecessor-version":[{"id":203,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/197\/revisions\/203"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}