J. Serra

activity

20112023

most citedBoundary regularity for fully nonlinear integro-differential equations

77 citations · 190 across the 25 of their papers we have counts for

collaborators

38 papers

math.DG2023

Nonlocal approximation of minimal surfaces: optimal estimates from stability

Hardy Chan, Serena Dipierro, Joaquim Serra +1

Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh…

math.AP2023★ 2 cited

Regularity theory for nonlocal obstacle problems with critical and subcritical scaling

Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, t…

math.DG2023

Yau's conjecture for nonlocal minimal surfaces

Michele Caselli, Enric Florit-Simon, Joaquim Serra

We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$ -dimensional Riemannian manifold we construct infinitel…

math.AP2021★ 4 cited

Free boundary partial regularity in the thin obstacle problem

Federico Franceschini, Joaquim Serra

For the thin obstacle problem in $R^{n}$ , $n \geq 2$ , we prove that at all free boundary points, with the exception of a $(n - 3)$ -dimensional set, the solution differs from it…

math.AP2021★ 3 cited

Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime

Xavier Cabre, Eleonora Cinti, Joaquim Serra

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(- Δ)^{s /2} u = u - u^{3}$ , $∣ u ∣ < 1$ in $R^{n}$ . For every $s \in (0, 1)$ and dimension $n \geq 2$ , we est…

math.AP2021

Interface regularity for semilinear one-phase problems

Alessandro Audrito, Joaquim Serra

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Ber…

math.AP2021

The singular set in the Stefan problem

Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n - 1$ . - The solution…

math.AP2020★ 1 cited

Non-symmetric stable operators: regularity theory and integration by parts

Serena Dipierro, Xavier Ros-Oton, Joaquim Serra +1

We study solutions to $Lu = f$ in $Ω \subset R^{n}$ , being $L$ the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solu…

math.AP2020

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

Eleonora Cinti, Federico Glaudo, Aldo Pratelli +2

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homog…

math.AP2019

Generic regularity of free boundaries for the obstacle problem

Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $R^{n}$ . By classical results of Caffarelli, the free boundary is $C^…

math.AP2019

Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Xavier Cabre, Alessio Figalli, Xavier Ros-Oton +1

In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n \leq 9$ . This result,…

math.AP2018

Regularity of minimal surfaces with lower dimensional obstacles

Xavier Fernández-Real, Joaquim Serra

We study the Plateau problem with a lower dimensional obstacle in $R^{n}$ . Intuitively, in $R^{3}$ this corresponds to a soap film (spanning a given contour) that is…

math.AP2017★ 4 cited

Stable $s$ -minimal cones in $R^{3}$ are flat for $s \sim 1$

Xavier Cabre, Eleonora Cinti, Joaquim Serra

We prove that half spaces are the only stable nonlocal $s$ -minimal cones in $R^{3}$ , for $s \in (0, 1)$ sufficiently close to $1$ . This is the first classification result of st…

math.AP2017

On the fine structure of the free boundary for the classical obstacle problem

Alessio Figalli, Joaquim Serra

In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular po…

math.AP2017★ 13 cited

Quantitative stability of the free boundary in the obstacle problem

Sylvia Serfaty, Joaquim Serra

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $R^{n}$ ( $n \geq 2$ ) under perturbations of th…

math.AP2017

On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

Alessio Figalli, Joaquim Serra

We prove that every bounded stable solution of \[ (-Δ)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u (x) = ϕ (e \cdot x)$ for some $e \in S^{2}$ , wher…

math.AP2016

Improvement of flatness for nonlocal phase transitions

Serena Dipierro, Joaquim Serra, Enrico Valdinoci

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(- Δ)^{s /2} u = u - u^{3}$ , with~ $s \in (0, 1)$ , we obtain a res…

math.AP2016

Nonlocal phase transitions: rigidity results and anisotropic geometry

Serena Dipierro, Joaquim Serra, Enrico Valdinoci

We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form $(- Δ)^{s /2} u = u - u^{3},$ with~ $s \in (0, 1)$ .…

math.AP2016★ 1 cited

The boundary Harnack principle for nonlocal elliptic operators in non-divergence form

Xavier Ros-Oton, Joaquim Serra

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if…

math.AP2016

The structure of the free boundary in the fully nonlinear thin obstacle problem

Xavier Ros-Oton, Joaquim Serra

We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is $C^{1}$ near regular points.

math.AP2016

Quantitative flatness results and $B V$ -estimates for stable nonlocal minimal surfaces

Eleonora Cinti, Joaquim Serra, Enrico Valdinoci

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$ -fractional perimeter as a particular case. On the one han…

math.AP2016★ 70 cited

Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries

Luis Caffarelli, Xavier Ros-Oton, Joaquim Serra

We study the obstacle problem for integro-differential operators of order $2 s$ , with $s \in (0, 1)$ . Our main result establishes that the free boundary is $C^{1, γ}$ and $u\in C^{1,s}…

math.AP2015★ 1 cited

Boundary regularity estimates for nonlocal elliptic equations in $C^{1}$ and $C^{1, α}$ domains

Xavier Ros-Oton, Joaquim Serra

We establish sharp boundary regularity estimates in $C^{1}$ and $C^{1, α}$ domains for nonlocal problems of the form $Lu = f$ in $Ω$ , $u = 0$ in $Ω^{c}$ . Here, $L$ is a nonlocal elliptic op…

math.AP2015

An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

Xavier Cabre, Joaquim Serra

We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are i…

Nonlocal approximation of minimal surfaces: optimal estimates from stability

Regularity theory for nonlocal obstacle problems with critical and subcritical scaling

Yau's conjecture for nonlocal minimal surfaces

Free boundary partial regularity in the thin obstacle problem

Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime

Interface regularity for semilinear one-phase problems

The singular set in the Stefan problem

Non-symmetric stable operators: regularity theory and integration by parts

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

Generic regularity of free boundaries for the obstacle problem

Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Regularity of minimal surfaces with lower dimensional obstacles

Stable s-minimal cones in R3 are flat for s∼1

On the fine structure of the free boundary for the classical obstacle problem

Quantitative stability of the free boundary in the obstacle problem

On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

Improvement of flatness for nonlocal phase transitions

Nonlocal phase transitions: rigidity results and anisotropic geometry

The boundary Harnack principle for nonlocal elliptic operators in non-divergence form

The structure of the free boundary in the fully nonlinear thin obstacle problem

Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces

Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries

Boundary regularity estimates for nonlocal elliptic equations in C1 and C1,α domains

An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

Stable $s$ -minimal cones in $R^{3}$ are flat for $s \sim 1$

Quantitative flatness results and $B V$ -estimates for stable nonlocal minimal surfaces

Boundary regularity estimates for nonlocal elliptic equations in $C^{1}$ and $C^{1, α}$ domains