Tesi di laurea specialistica
A class of existence results for the singular Liouville equation
SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di studi
relatore Prof. Malchiodi, Andrea
- Min-max Schemes
- Variational Methods
- Geometric PDEs
- Blow-up analysis
Data inizio appello
Data di rilascio
We consider a class of elliptic PDEs on closed surfaces with exponential nonliearities and Dirac deltas on the right-hand side. The study arises from abelian Chern-Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities.<br><br>A general existence result is proved using global variational methods and some examples of application are studied in detail. The analytic problem is reduced to a topological problem concerning the contractibility of a model space associated to the equation, the so-called space of formal baricenters. Finally, a conjecture is presented in order to reduce such topological investigation to test a very simple algebraic relation involving the parameters that appear in the equation.
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