arXiv subjects
Publications and source records attributed to Daniela De Silva.
We prove an analogue for a one-phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy-minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound.
We consider a one-phase free boundary problem with variable coefficients and non-zero right hand side. We prove that flat free boundaries are $C^{1,α}$ using a different approach than the classical supconvolution method of Caffarelli. We use this result to obtain that Lipschitz free boundaries are $C^{1,α}$.
We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.
We investigate regularity of minimizers in two dimensions for certain classes of non-smooth convex functionals. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others
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 $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in \cite{tz}, where global well-posedness was established for any $s>{4/9}$. 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.
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
We prove an a-priori bound for the Lipschitz constant of a smooth one-phase free boundary graph F(u) in two dimensions. The function u satisfies a fully nonlinear elliptic equation in its positive side, and the gradient of u is equal to 1 on F(u).
We establish that the initial value problem for the quadratic non-linear Schrödinger equation $$ iu_t - Δu = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this problem is $s_c=-1$ and previous work in \cite{c1} established local well-posedness for $s > -3/4$.
The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({\Bbb R^{n}})$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the ``$I$-method'' combined with a local in time Morawetz estimate.
The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $\Bbb T^{d}_λ = \Bbb R^{d}/{λ\Bbb Z^{d}}$, $d=1,2$.