Daniela De Silva

A singular energy minimizing free boundary

Abstract We consider the problem of minimizing the energy functional ∫(|∇u|2 + χ {u>0}). We show that the singular axissymmetric critical point of the functional is an energy minimizer in dimension 7. This is the first example of a non-smooth energy minimizer. It is analogous to the Simons cone, a least area hypersurface in dimension 8.

Free boundary regularity for a problem with right hand side

We consider a one-phase free boundary problem with variable coefficients and non-zero right hand side. We prove that flat free boundaries are

Boundary Harnack estimates in slit domains and applications to thin free boundary problems

C^{1,\alpha}

Global Well-Posedness and Polynomial Bounds for the Defocusing L 2-Critical Nonlinear Schrödinger Equation in ℝ

using a different approach than the classical supconvolution method of Caffarelli. We use this result to obtain that Lipschitz free boundaries are

Two-phase problems with distributed sources: regularity of the free boundary

C^{1,\alpha}

Minimizers of convex functionals arising in random surfaces

.

Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation

We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the

Regularity of Lipschitz free boundaries for the thin one-phase problem

C^\infty

Obstacle-type problems for minimal surfaces

regularity of the free boundary in the Signorini problem near non-degenerate points.

Existence and regularity of monotone solutions to a free boundary problem

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space H s (ℝ) for any . This improves the result in [25], where global well-posedness was established for any . We use the I-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution Iu. As a byproduct of our proof we also obtain that the H s norm of the solution obeys polynomial-in-time bounds.