Serena Dipierro, Xavier Ros‐Oton, Joaquim Serra, Enrico Valdinoci,
We study solutions to Lu=f in Ω⊂Rn, being L the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to Lu=f in Ω, u=0 in Ωc, in C1,α domains Ω. We show that solutions u satisfy u/dγ∈Cε∘(Ω‾), where d is the distance to ∂Ω, and γ=γ(L,ν) is an explicit exponent that depends on the Fourier symbol of operator L and on the unit normal ν to the boundary ∂Ω. On the other hand, we establish new integration by parts identities in half spaces for ...
Tópico(s): Stability and Controllability of Differential Equations
2022 - Elsevier BV | Advances in Mathematics
Serena Dipierro, Joaquim Serra, Enrico Valdinoci,
We establish an improvement of flatness result for critical points of Ginzburg-Landau energies with long-range interactions. It applies in particular to solutions of $(-\Delta)^{s/2}u=u-u^3$ in $\Bbb{R}^n$ with $s\in(0,1)$. As a corollary, we establish that solutions with asymptotically flat level sets are $1$D and prove the analogue of the De Giorgi conjecture (in the setting of minimizers) in dimension $n=3$ for all $s\in(0,1)$ and in dimensions $4\le n\le 8$ for $s\in(0,1)$ sufficiently close to $1$. The robustness ...
Tópico(s): Advanced Mathematical Modeling in Engineering
2020 - Johns Hopkins University Press | American Journal of Mathematics
Xavier Cabré, Alessio Figalli, Xavier Ros‐Oton, Joaquim Serra,
decreasing non-linearities . . . .
Tópico(s): Advanced Harmonic Analysis Research
2020 - Mittag-Leffler Institute | Acta Mathematica
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 $\mathbf {R}^{n}$ . By classical results of Caffarelli, the free boundary is $C^{\infty }$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$ -dimensional—that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $\mathcal{H}^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). ...
Tópico(s): Advanced Mathematical Modeling in Engineering
2020 - Springer Nature | Publications mathématiques de l IHÉS
Alessio Figalli, Joaquim Serra,
We prove that every bounded stable solution of $$\begin{aligned} (-\Delta )^{1/2} u + f(u) =0 \qquad \text{ in } \mathbb {R}^3 \end{aligned}$$is a 1D profile, i.e., $$u(x)= \phi (e\cdot x)$$ for some $$e\in {\mathbb {S}}^2$$, where $$\phi :\mathbb {R}\rightarrow \mathbb {R}$$ is a nondecreasing bounded stable solution in dimension one. Equivalently, stable critical points of boundary reaction problems in $$\mathbb {R}^{d+1}_+=\mathbb {R}^{d+1}\cap \{x_{d+1}\ge 0\}$$ of the form $$\begin{aligned} \int _{\{x_{d+1\ge 0}\}} \frac{1}{2} |\nabla U|^2 \,dx\, ...
Tópico(s): Geometric Analysis and Curvature Flows
2019 - Springer Science+Business Media | Inventiones mathematicae
Eleonora Cinti, Joaquim Serra, Enrico Valdinoci,
We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $n \geqslant 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_{1/2}$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) ...
Tópico(s): Geometric Analysis and Curvature Flows
2019 - Lehigh University | Journal of Differential Geometry
Alessio Figalli, Joaquim Serra,
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers (Caffarelli in Acta Math 139:155–184, 1977; J Fourier Anal Appl 4:383–402, 1998), regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $$C^1$$ manifolds of varying dimension. In two dimensions, this $$C^1$$ result has been improved to $$C^{1,\alpha }$$ by Weiss (Invent Math 138:23–50, 1999). In this paper ...
Tópico(s): Geometry and complex manifolds
2018 - Springer Science+Business Media | Inventiones mathematicae
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 Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 ∖Ω, and u1,u2 ≥ 0 in ℝn, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Hölder continuous up to the boundary in B1/2. These results will be used in forthcoming works on obstacle-type problems for nonlocal ...
Tópico(s): Differential Equations and Boundary Problems
2018 - Springer Science+Business Media | Potential Analysis
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 C1 near any regular point. This extends to the fully nonlinear setting the celebrated result of Athanasopoulos–Caffarelli–Salsa [1]. The proofs we present here are completely independent from those in [1], and do not rely on any monotonicity formula. Furthermore, an interesting and novel feature of our proofs is that we establish the regularity of the free ...
Tópico(s): Geometric Analysis and Curvature Flows
2017 - Elsevier BV | Advances in Mathematics
Xavier Ros‐Oton, Joaquim Serra, Enrico Valdinoci,
We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential operators of order 2s, with s∈(0,1). These identities involve local boundary terms, in which the quantity plays the role that ∂u∕∂ν plays in the second-order case. Here, u is any solution to Lu = f(x,u) in Ω, with u = 0 in ℝn∖Ω, and d is the distance to ∂Ω.
Tópico(s): Numerical methods in inverse problems
2017 - Taylor & Francis | Communications in Partial Differential Equations
Xavier Ros‐Oton, Joaquim Serra,
We establish sharp boundary regularity estimates in $$C^1$$ and $$C^{1,\alpha }$$ domains for nonlocal problems of the form $$Lu=f$$ in $$\Omega $$ , $$u=0$$ in $$\Omega ^c$$ . Here, L is a nonlocal elliptic operator of order 2s, with $$s\in (0,1)$$ . First, in $$C^{1,\alpha }$$ domains we show that all solutions u are $$C^s$$ up to the boundary and that $$u/d^s\in C^\alpha (\overline{\Omega })$$ , where d is the distance to $$\partial \Omega $$ . In $$C^1$$ domains, solutions are in general not comparable to $$d^s$$ , and we prove a boundary Harnack principle ...
Tópico(s): Advanced Mathematical Modeling in Engineering
2017 - Springer Science+Business Media | Annali di Matematica Pura ed Applicata (1923 -)
... found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with ...
Tópico(s): Geometric Analysis and Curvature Flows
2017 - Springer Science+Business Media | Chinese Annals of Mathematics Series B
Tópico(s): Nonlinear Differential Equations Analysis
2016 - Elsevier BV | Nonlinear Analysis
Luis Caffarelli, Xavier Ros‐Oton, Joaquim Serra,
Tópico(s): Numerical methods in inverse problems
2016 - Springer Science+Business Media | Inventiones mathematicae
Xavier Cabré, Xavier Ros‐Oton, Joaquim Serra,
Given an arbitrary convex cone of \mathbb R^n , we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex cone. This leads to isoperimetric inequalities with the optimal constant that were unknown even for a sector of the plane. Our result applies to all nonnegative homogeneous weights in \mathbb R^n satisfying a concavity condition in the cone. The condition is equivalent to ...
Tópico(s): Optimization and Variational Analysis
2016 - European Mathematical Society | Journal of the European Mathematical Society
Xavier Ros‐Oton, Joaquim Serra,
We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s∈(0,1). We consider the class of nonlocal operators L∗⊂L0, which consists of infinitesimal generators of stable Lévy processes belonging to the class L0 of Caffarelli–Silvestre. For fully nonlinear operators I elliptic with respect to L∗, we prove that solutions to Iu=f in Ω, u=0 in Rn∖Ω, satisfy u/ds∈Cs+γ(Ω¯), where d is the distance to ∂Ω and f∈Cγ. We expect the ...
Tópico(s): Differential Equations and Boundary Problems
2016 - Duke University Press | Duke Mathematical Journal
Xavier Ros‐Oton, Joaquim Serra,
We establish sharp regularity estimates for solutions to Lu=f in Ω⊂Rn, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on Sn−1, called the spectral measure. First, we study the interior regularity of solutions to Lu=f in B1. We prove that if f is Cα then u belong to Cα+2s whenever α+2s is not an integer. In case f∈L∞, we show that the solution u is C2s when s≠1/2, and C2s−ϵ for all ϵ>0 when s=1/2. Then, we study the boundary regularity ...
Tópico(s): Stochastic processes and financial applications
2016 - Elsevier BV | Journal of Differential Equations
We establish $$C^{\sigma +\alpha }$$ interior estimates for concave nonlocal fully nonlinear equations of order $$\sigma \in (0,2)$$ with rough kernels. Namely, we prove that if $$u\in C^{\alpha }(\mathbb {R}^n)$$ solves in $$B_1$$ a concave translation invariant equation with kernels in $$\mathcal L_0(\sigma )$$ , then u belongs to $$C^{\sigma +\alpha }(\overline{B_{1/2}})$$ , with an estimate. More generally, our results allow the equation to depend on x in a $$C^\alpha $$ fashion. Our method of proof combines a Liouville theorem and a ...
Tópico(s): Advanced Harmonic Analysis Research
2015 - Springer Science+Business Media | Calculus of Variations and Partial Differential Equations
Xavier Ros‐Oton, Joaquim Serra,
Abstract We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| n+σ is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case a ij ≡ 0 and for σ = 2 in case that (a ij ) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for L in star-shaped domains with ...
Tópico(s): Nonlinear Differential Equations Analysis
2014 - Taylor & Francis | Communications in Partial Differential Equations
We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $$u_t = \mathrm{I}u$$ , where $$\mathrm{I}$$ is translation invariant and elliptic with respect to the class $$\mathcal L_0(\sigma )$$ of Caffarelli and Silvestre, $$\sigma \in (0,2)$$ being the order of $$\mathrm{I}$$ . We prove that if $$u$$ is a viscosity solution in $$B_1 \times (-1,0]$$ which is merely bounded in $$\mathbb {R}^n \times (-1,0]$$ , then $$u$$ is $$C^\beta $$ in space ...
Tópico(s): Numerical methods in inverse problems
2014 - Springer Science+Business Media | Calculus of Variations and Partial Differential Equations
Xavier Ros‐Oton, Joaquim Serra,
In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$ in $${\Omega, u\equiv0}$$ in $${{\mathbb R}^n\backslash\Omega}$$ . Here, $${s\in(0,1)}$$ , (−Δ) s is the fractional Laplacian in $${\mathbb{R}^n}$$ , and Ω is a bounded C 1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then $${u/\delta^s|_{\Omega}}$$ is C α up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$ ...
Tópico(s): Nonlinear Differential Equations Analysis
2014 - Springer Science+Business Media | Archive for Rational Mechanics and Analysis
Xavier Ros‐Oton, Joaquim Serra,
We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$.We also obtain the Pohozaev identity for this operator.Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$. As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\Delta)^s\phi=\lambda\phi$ in $\Omega$, $\phi\equiv0$ in $\mathbb{R}^n\setminus\Omega$.
Tópico(s): Numerical methods in inverse problems
2014 - American Institute of Mathematical Sciences | Discrete and Continuous Dynamical Systems
Xavier Ros‐Oton, Joaquim Serra,
We study the extremal solution for the problem $$(-\Delta )^s u=\lambda f(u)$$ in $$\Omega $$ , $$u\equiv 0$$ in $$\mathbb R ^n\setminus \Omega $$ , where $$\lambda >0$$ is a parameter and $$s\in (0,1)$$ . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $$n<4s$$ . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $$n< ...
Tópico(s): Numerical methods in inverse problems
2013 - Springer Science+Business Media | Calculus of Variations and Partial Differential Equations
Xavier Ros‐Oton, Joaquim Serra,
We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of (−Δ)su=g in Ω, u≡0 in Rn\Ω, for some s∈(0,1) and g∈L∞(Ω), then u is Cs(Rn) and u/δs|Ω is Cα up to the boundary ∂Ω for some α∈(0,1), where δ(x)=dist(x,∂Ω). For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Hölder estimates for u and u/δs. Namely, the Cβ ...
Tópico(s): Nonlinear Differential Equations Analysis
2013 - Elsevier BV | Journal de Mathématiques Pures et Appliquées
Juan Carlos Ruiz‐Rodríguez, Adolf Ruiz-Sanmartín, Vicent Ribas, Jesús Caballero, Alejandra García-Roche, Jordi Riera, Xavier Nuvials, Miriam de Nadal, Oriol Solà-Morales, Joaquim Serra, Jordi Rello,
Tópico(s): Heart Rate Variability and Autonomic Control
2013 - Springer Science+Business Media | Intensive Care Medicine
Jordi Rello, Marcos Pérez-Carrasco, Oriol Roca, Garyphallia Poulakou, Jéssica Romualdo Souto, César Laborda, Joan Balcells, Joaquim Serra, Joan Ramón Masclans,
The experience with high-flow nasal cannula (HFNC) oxygen therapy in severe acute respiratory infection (SARI) is limited. The objective was to assess the effectiveness of HFNC oxygen therapy in adult patients with SARI by confirmed 2009 influenza A/H1N1v infection (by real-time reverse transcription polymerase chain reaction testing). A single-center post hoc analysis of a cohort of intensive care unit patients admitted with SARI due to 2009 Influenza A/H1N1v was done. High-flow nasal cannula (Optiflow; ...
Tópico(s): Pneumonia and Respiratory Infections
2012 - Elsevier BV | Journal of Critical Care
Xavier Ros‐Oton, Joaquim Serra,
In this Note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped domains. Dans cette Note, nous présentons lʼidentité de Pohozaev pour le Laplacien fractionnaire. Comme conséquence de cette identité, nous prouvons la non-existence de solutions non triviales pour les problèmes semi-linéaires avec nonlinéarité sur-critique dans ...
Tópico(s): Nonlinear Differential Equations Analysis
2012 - Elsevier BV | Comptes Rendus Mathématique
Xavier Cabré, Xavier Ros‐Oton, Joaquim Serra,
In this Note we present the solution of some isoperimetric problems in open convex cones of Rn in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990. Dans ...
Tópico(s): Nonlinear Partial Differential Equations
2012 - Elsevier BV | Comptes Rendus Mathématique