Andrea Malchiodi, SISSA

Abstract: The minicourse will cover some aspects in the study of nonlinear elliptic equations on surfaces or bidimensional domains with exponential nonlinearities:

1) I will review the basic notions of Gaussian Curvature and Laplace-Beltrami operator, and I will treat the Poincaré Uniformization Theorem.

2) Then I will discuss the Nirenberg problem on S^2 (namely prescibing the Gaussian curvature via conformal transformations).

3) Finally, I will treat blow-up analysis of solutions in domains and, time permitting, discuss existence techniques for nonlinear scalar field equations on surfaces.

Abstract:	The minicourse will cover some aspects in the study of nonlinear elliptic equations on surfaces or bidimensional domains with exponential nonlinearities:

	1) I will review the basic notions of Gaussian Curvature and Laplace-Beltrami operator, and I will treat the Poincaré Uniformization Theorem.

	2) Then I will discuss the Nirenberg problem on S^2 (namely prescibing the Gaussian curvature via conformal transformations).

	3) Finally, I will treat blow-up analysis of solutions in domains and, time permitting, discuss existence techniques for nonlinear scalar field equations on surfaces.