Integro-differential equations: regularity theory and Pohozaev identities

• Autores: Xavier Ros
• Directores de la Tesis: Xavier Cabré Vilagut (dir. tes.)
• Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2014
• Idioma: inglés
• Tribunal Calificador de la Tesis: Susanna Terracini (presid.), Juan de la Cruz de Solà-Morales i Rubio (secret.), Alessio Figalli (voc.), Jean-Michel Roquejoffre (voc.), José Antonio Carrillo (voc.)
• Materias:
  • Matemáticas
    • Análisis y análisis funcional
      • Calculo de variaciones
      • Convexidad, desigualdades
      • Ecuaciones integrales
      • Ecuaciones diferenciales en derivadas parciales
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • The main topic of the thesis is the study of Elliptic PDEs. It is divided into three parts: (I) integro-differential equations, (II) stable solutions to reaction-diffusion problems, and (III) weighted isoperimetric and Sobolev inequalities. Integro-differential equations arise naturally in the study of stochastic processes with jumps, and are used in Finance, Physics, or Ecology. The most canonical example of integro-differential operator is the fractional Laplacian (the infinitesimal generator of the radially symmetric stable process). In the first Part of the thesis we find and prove the Pohozaev identity for such operator. We also obtain boundary regularity results for general integro-differential operators, as explained next. In the classical case of the Laplacian, the Pohozaev identity applies to any solution of linear or semilinear problems in bounded domains, and is a very important tool in the study of elliptic PDEs. Before our work, a Pohozaev identity for the fractional Laplacian was not known. It was not even known which form should it have, if any. In this thesis we find and establish such identity. Quite surprisingly, it involves a local boundary term, even though the operator is nonlocal. The proof of the identity requires fine boundary regularity properties of solutions, that we also establish here. Our boundary regularity results apply to fully nonlinear integro-differential equations, but they improve the best known ones even for linear ones. Our work in Part II concerns the regularity of local minimizers to some elliptic equations, a classical problem in the Calculus of Variations. More precisely, we study the regularity of stable solutions to reaction-diffusion problems in bounded domains. It is a long standing open problem to prove that all stable solutions are bounded, and thus regular, in dimensions n<10. In dimensions n>=10 there are examples of singular stable solutions. The question is still open in dimensions 4