## 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.

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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