{"id":115826,"date":"2018-03-11T10:44:30","date_gmt":"2018-03-11T10:44:30","guid":{"rendered":"https:\/\/www.deberes.net\/tesis\/sin-categoria\/elliptic-and-parabolic-pdes-regularity-for-nonlocal-diffusion-equations-and-two-isoperimetric-problems\/"},"modified":"2018-03-11T10:44:30","modified_gmt":"2018-03-11T10:44:30","slug":"elliptic-and-parabolic-pdes-regularity-for-nonlocal-diffusion-equations-and-two-isoperimetric-problems","status":"publish","type":"post","link":"https:\/\/www.deberes.net\/tesis\/ecuaciones-diferenciales-en-derivadas-parciales\/elliptic-and-parabolic-pdes-regularity-for-nonlocal-diffusion-equations-and-two-isoperimetric-problems\/","title":{"rendered":"Elliptic and parabolic pdes: regularity for nonlocal diffusion equations and two isoperimetric problems"},"content":{"rendered":"

Tesis doctoral de Joaquim Serra Montol\u00ed <\/strong><\/h2>\n

The thesis is divided into two parts. the first part is mainly concerned with regularity issues for integro-differential (or nonlocal) elliptic and parabolic equations. In the same way that densities of particles with brownian motion solve second order elliptic or parabolic equations, densities of particles with l\u00e9vy diffusion satisfy these more general nonlocal equations. In this context, fully nonlinear nonlocal equations arise in stochastic control problems or differential games. The typical example of elliptic nonlocal operator is the fractional laplacian, which is the only translation, rotation and scaling invariant nonlocal elliptic operator. There many classical regularity results for the fractional laplacian —whose \u00abinverse’ is the riesz potential. For instance, the explicit poisson kernel for a ball is an \u00abold’ result, as well as the linear solvability theory in l^p spaces. however, very little was known on boundary regularity for these problems. A main topic of this thesis is the study of this boundary regularity, which is qualitatively very different from that for second order equations. We stablish a new boundary regularity theory for fully nonlinear (and linear) elliptic integro-differential equations. Our proofs require a combination of original techniques and appropriate versions of classical ones for second order equations (such as krylov\u00c2\u00bfs method). We also obtain new interior regularity results for fully nonlinear parabolic nonlocal equation with rough kernels. To do it, we develop a blow up and compactness method for viscosity solutions to fully nonlinear equations that allows us to prove regularity from liouville type theorems.This method is a main contribution of the thesis. The new boundary regularity results mentioned above are crucially used in the proof of another main result of the thesis: the pohozaev identity for the fractional laplacian. This identity is has a flavor of integration by parts formula for the fractional laplacian, with the important novely there appears a local boundary term (this was unusual with nolocal equations). in the second part of the thesis we give two instances of interaction between isoperimetry and partial differential equations. In the first one we use the alexandrov-bakelman-pucci method for elliptic pde to obtain new sharp isoperimetric inequalities in cones with densities by generalizing a proof of the classical isoperimetric inequality due to cabr\u00e9. Our new results contain as particular cases the classical wulff inequality and the isoperimetric inequality in cones of lions and pacella. In the second instance we use the isoperimetric inequality and the classical pohozaev identity to establish a radial symmetry result for second order reaction-diffusion equations. The novelty here is to include discontinuous nonlinearities. For this, we extend a two-dimensional argument of p.-L. Lions from 1981 to obtain now results in higher dimensions.<\/p>\n

 <\/p>\n

Datos acad\u00e9micos de la tesis doctoral \u00abElliptic and parabolic pdes: regularity for nonlocal diffusion equations and two isoperimetric problems<\/strong>\u00ab<\/h3>\n
    \n
  • T\u00edtulo de la tesis:<\/strong>\u00a0 Elliptic and parabolic pdes: regularity for nonlocal diffusion equations and two isoperimetric problems <\/li>\n
  • Autor:<\/strong>\u00a0 Joaquim Serra Montol\u00ed <\/li>\n
  • Universidad:<\/strong>\u00a0 Polit\u00e9cnica de catalunya<\/li>\n
  • Fecha de lectura de la tesis:<\/strong>\u00a0 17\/06\/2014<\/li>\n<\/ul>\n

     <\/p>\n

    Direcci\u00f3n y tribunal<\/h3>\n
      \n
    • Director de la tesis<\/strong>\n
        \n
      • Xavier Cabr\u00e9 Vilagut<\/li>\n<\/ul>\n<\/li>\n
      • Tribunal<\/strong>\n
          \n
        • Presidente del tribunal: Juan Luis V\u00e1zquez su\u00e1rez <\/li>\n
        • sylvia Serfaty (vocal)<\/li>\n
        • scott n. Armstrong (vocal)<\/li>\n
        • Javier Tolsa domenech (vocal)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

           <\/p>\n","protected":false},"excerpt":{"rendered":"

          Tesis doctoral de Joaquim Serra Montol\u00ed The thesis is divided into two parts. the first part is mainly concerned with […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"footnotes":""},"categories":[42946,32592,3185,19586,15596],"tags":[193139,228946,142506,228948,228947,75121],"class_list":["post-115826","post","type-post","status-publish","format-standard","hentry","category-calculo-de-variaciones","category-convexidad-y-desigualdades","category-ecuaciones-diferenciales-en-derivadas-parciales","category-ecuaciones-integrales","category-politecnica-de-catalunya","tag-javier-tolsa-domenech","tag-joaquim-serra-montoli","tag-juan-luis-vazquez-suarez","tag-scott-n-armstrong","tag-sylvia-serfaty","tag-xavier-cabre-vilagut"],"_links":{"self":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/115826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/comments?post=115826"}],"version-history":[{"count":0,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/posts\/115826\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/media?parent=115826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/categories?post=115826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.deberes.net\/tesis\/wp-json\/wp\/v2\/tags?post=115826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}