Evolution families in the space of p-nonexpansive mappings and harmonic Loewner chains in R2

• Autores: Luis Benítez Babilonia
• Directores de la Tesis: Lázaro Raúl Felipe Parada (dir. tes.)
• Lectura: En la Centro de Investigación en Matemáticas (CIMAT) ( México ) en 2012
• Idioma: inglés
• Enlaces
  • Tesis en acceso abierto en: cimat.repositorioinstitucional.mx
• Resumen

  • The purpose of this work is to present a version of the Loewner theory, without the requirement of analyticity of the functions belonging to the chain and the evolution families or transition functions.

    This approach is made in two parts.

    In the first part, we introduce two new concepts of subordination in the class of harmonic univalent functions. Then, we introduce two new types of harmonic Loewner chains according to the already given concepts of subordination. A characterization for one type of these harmonic Loewner chains is presented. The compactness of this family of harmonic Loewner chains is proved. For the other type of harmonic Loewner chains an ordinary differential equation is established in a particular case.

    Actually, a Loewner theory for complex-valued harmonic functions, whose real and imaginary parts not necessarily satisfy the Cauchy-Riemann equations, is constructed.

    The second part is taken into consideration, because enforcing that the composition of two harmonic functions is harmonic, is a very restrictive condition. Therefore, we have obtained interesting, but not very “fruitful” results. In this part, we just study the evolution families or transition functions in the class of nonexpansive functions with respect to a certain metric. We shown that these evolution families can be obtained by solving an ordinary differential equation for a certain vector field. The concept of an infinitesimal generator for these evolution families is given. Some characteristics of such infinitesimal generators are established. The nonlinear resolvent for a kind of functions is treated. An ordinary differential equation, which is satisfied by these evolution families, is obtained with some additional assumptions.