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Tesi etd-06262018-214528


Thesis type
Tesi di laurea magistrale
Author
FRANZ, GIADA
URN
etd-06262018-214528
Title
Construction of alpha-harmonic maps between spheres
Struttura
MATEMATICA
Corso di studi
MATEMATICA
Commissione
relatore Prof. Malchiodi, Andrea
controrelatore Prof. Frigerio, Roberto
Parole chiave
  • harmonic maps
  • perturbation method
  • semilinear elliptic problem
Data inizio appello
13/07/2018;
Consultabilità
secretata d'ufficio
Riassunto analitico
A map between compact Riemannian manifolds is called harmonic if it is a critical point of the Dirichlet energy (the integral of the square norm of the gradient).
Unfortunately, the Dirichlet functional suffers from a lack of compactness which causes problems when we try to find harmonic maps.
To make up for this problem, Sacks and Uhlenbeck introduced a perturbed functional adding an exponent slightly greater than 1 inside the integral.

The goal of this thesis is to construct critical points of the perturbed functional which don't come out by a standard compactness argument (such as the Mountain Pass Theorem).
In detail we construct a 1-parameter family of pseudo-critical points (that are points in which the norm of the gradient of the functional is small) suitably gluing two harmonic maps and we look for critical points close to this family, with the aid of a Lyapunov-Schmidt reduction.
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