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Daniela De Silva

Publications and source records attributed to Daniela De Silva.

At least 37 records · Page 2Linked to original sources
arXiv2019

A short proof of Boundary Harnack Inequality

We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

math.AP↗
arXiv2019

Almost minimizers of the one-phase free boundary problem

We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions.

math.AP↗
arXiv2018

Thin One-Phase almost minimizers

We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a noninfinitesimal notion of viscosity solutions.

math.AP↗
arXiv2018

Segregated configurations involving the square root of the laplacian and their free boundaries

We study the local structure and the regularity of free boundaries of segregated critical configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren's monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known resultsfor the standard diffusion to some anomalous case.

math.AP↗
arXiv2018

Quasi-Harnack inequality

In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale $r_0$ they satisfy a Weak Harnack type estimate. We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.

math.AP↗
arXiv2012

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

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,Ω) = \int_Ω|\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad Ω\subset \R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\p Ω$. We prove that the free boundary $F(u)=\p_{\R^n}\{u>0\}$ of a minimizer $u$ has locally finite $\mathcal{H}^{n-1}$ measure and is a $C^{2,α}$ surface except on a small singular set of Hausdorff dimension $n-3$. We also obtain $C^{2,α}$ regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.

math.AP↗
arXiv2011

C^{2,α}$ regularity of flat free boundaries for the thin one-phase problem

We prove $C^{2,α}$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary problem in the context of the fractional Laplacian $(-Δ)^{1/2}$.

math.AP↗