Research Interests

My research interest is in partial differential equations. I am particularly interested in the regularity theory for free boundary/phase transition problems. Also, I am currently investigating well-posedness issues for certain nonlinear dispersive equations.

Papers:

• Some remarks on nonlinear elliptic equations and applications to Hamilton-Jacobi equations, (with C. Trombetti), C.R. Acad. Sci. Paris, t. 333, Serie I, p. 91-96, 2001
• Estimates for the gradient of solutions of elliptic equations in Orlicz-Sobolev spaces, Ricerche di Matematica, vol. LI, issue 1, p. 25-47, 2002
• A singular energy minimizing free boundary, (with D. Jerison) submitted     [.pdf]

Supporting files:

Subsolution property     [.nb] Strict subsolution property     [.nb] Supersolution properties     [.nb]

• Existence and regularity of monotone solutions to free boundary problems, submitted
• Global well-posedness for a periodic nonlinear Schr\"odinger equation in 1D and 2D, (with N. Pavlovic, G. Staffilani and N. Tzirakis) to appear in Discrete and Continuous Dynamical Systems
• Low regularity solutions for a 2D quadratic non-linear Schr\"odinger equation, (with I. Bejenaru) to appear Trans. of AMS
• Global Well-Posedness for the $L^2$-critical nonlinear Schr\"odinger equation in higher dimensions, (with N. Pavlovic, G. Staffilani and N. Tzirakis) to appear in Communication on Pure and Applied Analysis
• Bernstein-type techniques for 2D free boundary graphs, submitted
• Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$, (with N. Pavlovic, G. Staffilani and N. Tzirakis) submitted
• Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D, (with O. Savin) submitted
• Gradient bound for energy minimizing free boundary graphs, (with D. Jerison) in preparation
• Rearrangements and radial graphs of constant mean curvature in hyperbolic space, (with J. Spruck) in preparation