{"id":319,"date":"2025-07-08T14:06:12","date_gmt":"2025-07-08T13:06:12","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=319"},"modified":"2025-07-08T14:13:26","modified_gmt":"2025-07-08T13:13:26","slug":"mesura-harmonica","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/mesura-harmonica\/","title":{"rendered":"Mesura harm\u00f2nica"},"content":{"rendered":"\n<p>El problema de Dirichlet consisteix a trobar funcions harm\u00f2niques \\( u:\\Omega\\to\\mathbb R\\) en un domini \\(\\Omega\\) amb els valors de frontera prescrits \\(u\\equiv f\\). Quan la dada de frontera \u00e9s cont\u00ednua i el domini \u00e9s prou regular podem demostrar exist\u00e8ncia i unicitat de la soluci\u00f3 amb continu\u00eftat a la clausura del domini \\(\\overline\\Omega\\).<\/p>\n<p>Donat un punt \\(x\\in\\Omega\\), l\u2019aplicaci\u00f3 \\(f\\mapsto u(x)\\) \u00e9s cont\u00ednua i, pel teorema de representaci\u00f3 de Riemann, existeix una mesura \\(\\omega^x\\) suportada en \\(\\partial\\Omega\\) de manera que \\(u(x)=\\int f(y)\\, d\\omega^x(y)\\). Aquesta s\u2019anomena mesura harm\u00f2nica.<\/p>\n<p><strong>Possibles<\/strong> l\u00ednies de treball:<\/p>\n<ul>\n<li>Punt de vista d\u2019an\u00e0lisi funcional: teorema de representaci\u00f3 de Riesz i \u00fas d\u2019aquest en la definici\u00f3 rigorosa de mesura harm\u00f2nica. Mirarem d\u2019entendre le teorema de representaci\u00f3 de Riesz [4] i com s\u2019aplica per obtenir una definici\u00f3 rigorosa de mesura harm\u00f2nica en qualsevol domini fitat, via m\u00e8tode de Perron [2], aix\u00ed com maneres possibles d\u2019estendre la definici\u00f3 a altres contextos no fitats.<\/li>\n<li>Punt de vista f\u00edsic\/estoc\u00e0stic: el moviment Browni\u00e0 i la relaci\u00f3 amb la mesura harm\u00f2nica amb el teorema de Kakutani. Llegirem l\u2019article original [3] i el posarem en el context actual, mirant la formulaci\u00f3 exacta del resultat en relaci\u00f3 amb el m\u00e8tode de Perron [2].<\/li>\n<li>Punt de vista d\u2019an\u00e0lisi complexa i teoria geom\u00e8trica de la mesura: la dimensi\u00f3 de la mesura harm\u00f2nica al pla. Estudiarem el teorema dels germans Riesz i dimensi\u00f3 en conjunts generals. [1,2].<\/li>\n<li>Punt de vista d&#8217;EDPs: la mesura el\u00b7l\u00edptica. Mirarem com passar les definicions obtingudes per funcions harm\u00f2niques a funcions \\(L\\)-harm\u00f2niques on \\(L\\) \u00e9s un operador el\u00b7l\u00edptic en forma de diverg\u00e8ncia. [5]<\/li>\n<\/ul>\n<p>[1] Garnett JB, Marshall DE. <em>Harmonic Measure<\/em>. Cambridge University Press; 2005.<\/p>\n<p>[2] Prats, M. Tolsa, X. <em>Notes on harmonic measure<\/em> (preprint, 2023) &#8211; <a href=\"https:\/\/mat.uab.es\/~xtolsa\/mesuraharmonica.pdf\">https:\/\/mat.uab.es\/~xtolsa\/mesuraharmonica.pdf<\/a><\/p>\n<p>[3] Kakutani, S. (1944). <a href=\"https:\/\/doi.org\/10.3792%2Fpia%2F1195572742\">&#8220;On Brownian motion in n-space&#8221;<\/a>. Proc. Imp. Acad. Tokyo. <strong>20<\/strong> (9): <em>648\u2013652<\/em><\/p>\n<p>[4] Rudin, W. <em>Real and complex analysis<\/em>. Tata McGraw-Hill Education, 1987.<\/p>\n<p>[5] Heinonen, J., Kilpel\u00e4inen, T. and Martio, O. &#8220;Fine topology and quasilinear elliptic equations.&#8221; <i>Annales de l&#8217;institut Fourier<\/i>. Vol. 39. No. 2. 1989.<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>El problema de Dirichlet consisteix a trobar funcions harm\u00f2niques \\( u:\\Omega\\to\\mathbb R\\) en un domini \\(\\Omega\\) amb els valors de frontera prescrits \\(u\\equiv f\\). Quan la dada de frontera \u00e9s cont\u00ednua i el domini \u00e9s prou regular podem demostrar exist\u00e8ncia i unicitat de la soluci\u00f3 amb continu\u00eftat a la clausura del domini \\(\\overline\\Omega\\). Donat un [&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-319","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\/319","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=319"}],"version-history":[{"count":6,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/319\/revisions"}],"predecessor-version":[{"id":325,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/319\/revisions\/325"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}