Marcello LUCIA

September 24th, 2:00pm-3:30pm (in person)
Samuel Magill (CUNY Graduate Center)
Liouville Equation with Dirac Singularities
Abstract: I will present some recent results obtained for some Dirichlet semilinear problems with exponential nonlinearity Je^t and Dirac source terms located on a two dimensional domain. So far the existence of solutions was known when the parameter J is below 2 pi, and I will show how this can be improved up to some optimal threshold.

October 22nd, 2:00pm-3:30pm (in person)
Andras Balogh (College of Staten Island, CUNY)
Global Solutions of the Higgs Boson Equation in de Sitter Spacetime
Abstract: We investigate the Higgs Boson Equation in de Sitter Spacetime. We prove the existence and uniqueness of global–in–time solutions for initial data of arbitrary size with compact support. The proof uses Galerkin's method, based on various a priori norm estimates.

November 19th, 2:00pm-3:30pm (in person)
TBA
TBA
Abstract:

March 19th (in person):
Samuel Magill (CUNY Graduate Center)
Variational Sub-super method in some PDE problems with singularities
Abstract: The sub-super method to detect a solution to nonlinear problems has been reconsidered by Struwe in a variational context. I will discuss how it can be refined, and discuss how it can be applied to find some solutions for a class of problems that involve Dirac measures.

April 16th (in person):
Inigo Urtiaga Erneta (Rutgers University)
Regularity of oblique transmission problems
Abstract: I will discuss transmission problems model phenomena in domains made up of multiple adjacent phases. While a variational ``divergence-form'' theory for such problems is by now classical, a non-variational approach has only emerged recently. This talk concerns the regularity of viscosity solutions to such transmission problems in non-divergence form.
I will present new results in the case of flat interfaces, where the transmission condition is allowed to depend on both the normal and tangential derivatives of the solution. This condition can be interpreted as a nonlinear coupling of oblique derivatives from each side of the interface. Our main result establishes optimal piecewise Hölder regularity of viscosity solutions.

April 23rd, at 5:30pm (in person):
Thesis Defense of Samuel Magill
Variational methods for semilinear PDEs with Dirac Singularities
Abstract: We consider semilinear elliptic PDEs in two dimensions involving Dirac measures. Using a sub- and supersolution method designed for variational problems introduced by Struwe, we will present some new results for different classes of PDEs.
We will first cover the case of some PDEs with exponential nonlinearity subjected to Dirichlet boundary condition, and then discuss a broad class of problems on the torus that arise from some vortex condensation models. The method does not rely on a monotone iteration scheme or maximum principle arguments allowing it to extend naturally to coupled systems.