Research
Elliptic Partial Differential Equations arising in mathematical physics and geometry.
In particular:
Liouville equation
Super-Liouville equation
Toda system
Sinh-Gordon equation
Gelfand equation
Moser-Trudinger inequality
Free boundary problems
I am also part of the DEG1 math group.
Some expository seminars:
Free boundary problems
Uniqueness and monotonicity of solutions via bifurcation analysis. See for example (pdf).
Bifurcation theory
Bifurcation theory: study of the qualitative behavior of bifurcation diagrams. See for example (pdf).
Uniqueness and non-degeneracy of bubbling solutions
Uniqueness and non-degeneracy of bubbling solutions to Liouville-type problems around a given blow-up set: based on sharp profile estimates of bubbling solutions. See for example (pdf1, pdf2, pdf3).
Uniqueness of solutions
Symmetry and uniqueness of solutions to Liouville-type problems, both on bounded domains and on spheres. The arguments are based on the isoperimetric inequality and on the Sphere Covering Inequality which is used to compare different solutions to Liouville equations. See for example (pdf1, pdf2).
Super-Liouville
Super-Liouville problems: motivated by the supersymmetric extension of the Liouville theory. Exploiting Nehari manifolds jointly with either mountain pass / linking geometry or bifurcation analysis we provide existence results on compact surfaces. See for example (pdf1, pdf2).
Sinh-Gordon equation
Liouville-type equation arising in fluid dynamics and constant mean curvature surfaces. The existence issue is addressed by exploiting its variational structure and new improved Moser-Trudinger inequalities. As for the Toda system, the existence of solutions in a general setting is still open. Moreover, the blow-up analysis is performed yielding quantization/non-quantization of blow-up masses and exclusion of boundary blow-up points. However, the general blow-up picture has to be completed. Finally, some results concerning the Leray-Schauder degree are obtained. See for example (pdf1, pdf2, pdf3, pdf4 ).
Toda system
Liouville-type system arising in Gauge theory and in the description of holomorphic curves. The approach is based on its variational structure. We derive new improved Moser-Trudinger inequalities which allow to carry on min-max methods yielding existence of solutions by detecting a change of topology of sublevels of the associated functional. The existence issue in a general setting is still open. See for example (pdf1, pdf2).