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Daniela De Silva

Publications and source records attributed to Daniela De Silva.

At least 19 recordsLinked to original sources
arXiv2026

Viscosity and variational approaches in free boundary problems: A Volume in Honor of Sandro Salsa

Free boundary problems (FBPs) are differential equations where the domain depends on the solution, with applications in diverse fields such as flame propagation, financial mathematics, and tumor growth. Classical examples include the Bernoulli problem, the obstacle problem, and the Stefan problem. The field has advanced significantly through variational and geometric approaches, notably by Alt, Caffarelli, and collaborators, who developed monotonicity formulas and viscosity solutions to address free boundary regularity. Extensions have included nonlinear operators, inhomogeneous problems, and lower-dimensional cases like the thin Bernoulli problem. Evolution problems, such as the Hele-Shaw flow, also connect to FBPs. To celebrate these developments and honor Sandro Salsa's pioneering contributions, a special session on FBPs was held at the AMS-UMI joint meeting in Palermo (2024). This is the editorial of a volume dedicated to Sandro Salsa on the occasion of his 75th birthday.

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arXiv2025

The free boundary for a superlinear system

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_Ω\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0<p<1, $$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,α}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity.

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arXiv2024

Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones

We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set $K \subset \mathbb{R}^n$ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \cite{C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain $K$. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest.

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arXiv2022

Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

We investigate the rigidity of global minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2),$$ when the exponent $γ$ is close to the extremes of the admissible values. In particular we show that global minimizers in $\mathbb{R}^n$ are one-dimensional if $γ$ is close to 2 and $n \le 7$, or if $γ$ is close to $0$ and $n \le 4$.

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arXiv2022

Almost minimizers for a sublinear system with free boundary

We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(λ_+(x)|\mathbf{u}^+|^{q+1}+λ_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0 0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.

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arXiv2022

Uniform density estimates and $Γ$-convergence for the Alt-Phillips functional of negative powers

We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.

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arXiv2022

The Alt-Phillips functional for negative powers

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $$\int_Ω\left(\frac 1 2 |\nabla u|^2 + u^γ χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (-2,0),$$ and establish a $Γ$-convergence result of the rescaled energies to the perimeter functional as $γ\to -2$.

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arXiv2021

Almost minimizers for a singular system with free boundary

In this paper we study vector-valued almost minimizers of the energy functional $$ \int_D\left(|\nabla\mathbf{u}|^2+2|\mathbf{u}|\right)\,dx . $$ We establish the regularity for both minimizers and the "regular" part of the free boundary. The analysis of the free boundary is based on Weiss-type monotonicity formula and the epiperimetric inequality for the energy minimizers.

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arXiv2021

Inhomogeneous global minimizers to the one-phase free boundary problem

Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.

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arXiv2021

On the Parabolic Boundary Harnack Principle

We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.

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arXiv2020

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced in \cite{CRS} as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a "regular" and a "singular" part. Finally we use a viscosity approach to prove $C^{1,α}$ regularity of the regular part of the free boundary.

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arXiv2020

Perturbative estimates for the one-phase Stefan Problem

We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established by the first author.

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arXiv2019

On certain degenerate one-phase free boundary problems

We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more generally for minimizers of the Alt-Phillips functional.

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arXiv2019

Improvement of flatness for vector valued free boundary problems

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,α}$ regularity, as well-known in the scalar case \cite{AC,C2}. While in \cite{MTV2} the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of \cite{D}. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in \cite{DR, DSS}.

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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.

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