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BCAMSevero Ochoa Award

basque center for applied mathematics

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Eceizabarrena, Daniel

Thesis

  • The Pointwise Convergence of the Solution to the Schrödinger Equation

    Date:
    27-06-2016
    University:
    Universidad Autónoma de Madrid (UAM)
    Country:
    Spain
    File:
    Deskargatu PDF
    Comments:
    The present document corresponds to the subject Trabajo Fin de Máster being part of the Master's Degree in Mathematics and Applications at Universidad Autónoma de Madrid. The objective of this dissertation is to analyse the almost everywhere convergence of the solution to the Schrödinger equation to the given initial data.

    This problem was presented by Lennart Carleson in 1980, when he proved convergence for the one space dimensional problem and for data lying in Sobolev spaces Hs with exponents s ≥ 1/4 . Since then, many partial results have shown that Sobolev spaces are the convenient setup to work with. The problem was solved in one dimension in 1981 by B.E.J. Dahlberg and C.E. Kenig, who showed Carleson's condition was also necessary. We review both results in the first chapter.

    The question still remains unsolved for higher dimensions. Throughout the rest of the chapters, we present several partial results, particularly the best necessary and sufficient conditions known so far.


  • Distribution Theory and Fundamental Solutions of Differential Operators

    Date:
    24-06-2015
    University:
    Euskal Herriko Unibertsitatea (UPV/EHU)
    Country:
    Spain
    File:
    Deskargatu PDF
    Comments:
    This work is the written document of the subject Gradu Amaierako Lana, the final thesis corresponding to the Degree in Mathematics.

    The objective of this dissertation is to study the basics of the theory of distributions and show how it can be used to obtain what we call fundamental solutions of partial differential equations. The first two chapters are devoted to describe several spaces of test functions and distributions. The third chapter covers the Fourier transform, which gives a way to connect distributions and partial differential equations. In the last chapter we work to obtain some fundamental solutions, such as the ones for the heat equation, the Schrödinger equation, the Laplace equation and the Cauchy-Riemann equations.
Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del PaÌs Vasco (UPV/EHU)