Now showing items 1-10 of 10

  • Boundary regularity for the fractional heat equation 

    Fernández-Real Girona, Xavier (Universitat Politècnica de Catalunya, 2014-09)
    Bachelor thesis
    Open Access
    In this dissertation we present an introduction to nonlocal operators, and in particular, we study the fractional heat equation, which involves the fractional Laplacian of order 2s. In the first chapters we make a review ...
  • Ground states in Mathematical Physics 

    Doce Llisó, Edurne (Universitat Politècnica de Catalunya, 2015-09)
    Bachelor thesis
    Open Access
    The main aim of this bachelor's thesis is to introduce the concept of a ground state and prove its existence for the nonlinear diffusion equation $$ -\Delta u + a u = b |u|^\alpha u \ \ \text{in} \ \mathbb{R}^N, $$ as well ...
  • Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties 

    González Nogueras, María del Mar; Saéz, Mariel; Sire, Yannick (2014-12-01)
    Article
    Restricted access - publisher's policy
    We investigate the equation; (-Delta(Hn))(gamma) w = f(w) in H-n,; where (-Delta(Hn))(gamma) corresponds to the fractional Laplacian on hyperbolic space for gamma is an element of(0, 1) and f is a smooth nonlinearity that ...
  • Local integration by parts and Pohozaev indentities for higuer order fractional Laplacians 

    Ros Oton, Xavier; Serra Montolí, Joaquim (2015-05-01)
    Article
    Restricted access - publisher's policy
    We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian (-Delta)(s) with s > 1. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary ...
  • Nonexistence results for nonlocal equations with critical and supercritical nonlinearities 

    Ros Oton, Xavier; Serra Montolí, Joaquim (2015-01-02)
    Article
    Open Access
    We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form; [GRAPHICS]; These operators are infinitesimal generators of symmetric Levy processes. Our results ...
  • Periodic solutions to PDEs with fractional diffusion 

    Felipe Navarro, Juan Carlos (Universitat Politècnica de Catalunya, 2016-01)
    Bachelor thesis
    Open Access
    The aim of this Bachelor's Thesis is the study of periodic solutions to nonlinear equations involving the fractional Laplace operator. Our starting point is the Benjamin-Ono equation in water waves, a completely integrable ...
  • Positive solutions of nonlinear problems involving the square root of the Laplacian 

    Cabré Vilagut, Xavier; Tan, Jinggang (2009-05)
    External research report
    Open Access
    We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, ...
  • Some constructions for the fractional Laplacian on noncompact manifolds 

    Banica, Valeria; González Nogueras, María del Mar; Saéz, Mariel (2015-01-01)
    Article
    Open Access
    We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact ...
  • Stable and periodic solutions to nonlinear equations with fractional diffusion 

    Sanz Perela, Tomás (Universitat Politècnica de Catalunya, 2016-07)
    Master thesis
    Open Access
    The aim of this thesis is to study stable solutions to nonlinear elliptic equations involving the fractional Lapacian. More precisely, we study the extremal solution for the problem $(\Delta )^s u = \lambda f(u)$ in $\Omega$, ...
  • The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary 

    Ros Oton, Xavier; Serra Montolí, Joaquim (2014-03)
    Article
    Restricted access - publisher's policy
    We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (-d)su=g in O, u=0 in Rn\O, for some s¿(0, 1) and g¿L8(O), then u is Cs(Rn) ...