Searcharxiv⌕ Search

arXiv subjects

Xavier Ros-Oton

Publications and source records attributed to Xavier Ros-Oton.

At least 55 records · Page 3Linked to original sources
arXiv2019

Obstacle problems for integro-differential operators: Higher regularity of free boundaries

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in $C^{k,α}$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\infty$) and by the second author and Serra (for $k=1$).

math.AP↗
arXiv2018

On global solutions to semilinear elliptic equations related to the one-phase free boundary problem

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $Δu=f(u)$ in $\mathbb{R}^n$, where $f$ is smooth, non-negative, with support in the interval $[0,1]$. In such setting, any "blow-down" of the solution $u$ will converge to a global solution to the classical one-phase free boundary problem of Alt-Caffarelli. In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study here the 1D symmetry of solutions $u$ that are energy minimizers. Our main result establishes that, in dimensions $n<6$, if $u$ is axially symmetric and stable then it is 1D.

math.AP↗
arXiv2018

Higher-order boundary regularity estimates for nonlocal parabolic equations

We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\timesΩ$ where $I\subset\mathbb{R}$, $Ω\subset\mathbb{R}^n$ and $f$ is Hölder continuous. The nonlocal operators $L$ considered are those arising in stochastic processes with jumps such as the fractional Laplacian $(-Δ)^s$, $s\in(0,1)$. Our main result establishes that, if $f$ is $C^γ$ is space and $C^{γ/2s}$ in time, and $Ω$ is a $C^{2,γ}$ domain, then $u/d^s$ is $C^{s+γ}$ up to the boundary in space and $u$ is $C^{1+γ/2s}$ up the boundary in time, where $d$ is the distance to $\partialΩ$. This is the first higher order boundary regularity estimate for nonlocal parabolic equations, and is new even for the fractional Laplacian in $C^\infty$ domains.

math.AP↗
arXiv2017

Obstacle problems and free boundaries: an overview

Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author.

math.AP↗
arXiv2017

Boundary regularity, Pohozaev identities, and nonexistence results

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $Ω$, $u\equiv0$ in $\mathbb R^n\backslashΩ$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}. After this, we present the Pohozaev identities established in \cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integro-differential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(-Δ)^s$.

math.AP↗
arXiv2017

Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $φ$ satisfies $Δφ\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,α}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical Laplacian \cite{C-obst1,C-obst2}, and it is new even for the Signorini problem (which corresponds to the particular case of the $\frac12$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-φ)\geq c\,r^2$, valid at all free boundary points $x_0$.

math.AP↗
arXiv2017

Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.

math.AP↗
arXiv2017

Regularity theory for general stable operators: parabolic equations

We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times Ω$, with $I \subset \mathbb{R}$ and $Ω\subset\mathbb{R}^n$. The operators $L$ we consider are infinitessimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that $u$ is $C^{2s+α}$ in $x$ and $C^{1+\fracα{2s}}$ in $t$, whenever $f$ is $C^α$ in $x$ and $C^{\fracα{2s}}$ in $t$. In the case $f\in L^\infty$, we prove that $u$ is $C^{2s-ε}$ in $x$ and $C^{1-ε}$ in $t$, for any $ε> 0$. On the other hand, we study the boundary regularity of solutions in $C^{1,1}$ domains. We prove that for solutions $u$ to the Dirichlet problem the quotient $u/d^s$ is Hölder continuous in space and time up to the boundary $\partialΩ$, where $d$ is the distance to $\partialΩ$. This is new even when $L$ is the fractional Laplacian.

math.AP↗
arXiv2017

The obstacle problem for the fractional Laplacian with critical drift

We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{(-Δ)^s u + b \cdot \nabla u,\,u -φ\right\} = 0$ in $\mathbb{R}^n$, in the critical regime $s = \frac{1}{2}$. Our main result establishes the $C^{1,α}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \[ u(x)-φ(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tildeγ(x_0)} + o\left(|x-x_0|^{1+\tildeγ(x_0)+σ}\right), \] \[ \tildeγ(x_0) = \frac{1}{2}+\frac{1}π \arctan (b\cdot e), \] where $e \in \mathbb{S}^{n-1}$ is the normal vector to the free boundary, $σ>0$, and $c_0> 0$. We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent $\tildeγ(x_0)$ also depends on the operator.

math.AP↗
arXiv2017

On the regularity of the free boundary in the $p$-Laplacian obstacle problem

We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-Δ_p u,\,u-φ\bigr\}=0$ in $Ω\subset\mathbb R^n$. Here, $Δ_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\nabla φ\neq0$, the operator $Δ_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla φ=0$ then $Δ_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $\nabla φ=0$. On the one hand, for every $p\neq2$ we construct explicit global $2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not $C^1$ at points where $\nabla φ=0$. On the other hand, under the "concavity" assumption $|\nabla φ|^{2-p}Δ_p φ<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.

math.AP↗
arXiv2016

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

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,γ}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $x_0\in\partial\{u=φ\}$: (i) either $u(x)-φ(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+α})$ for some $c>0$, (ii) or $u(x)-φ(x)=o(|x-x_0|^{1+s+α})$, where $d$ is the distance to the contact set $\{u=φ\}$. Moreover, we show that the set of free boundary points $x_0$ satisfying (i) is open, and that the free boundary is $C^{1,γ}$ and $u\in C^{1,s}$ near those points. These results were only known for the fractional Laplacian \cite{CSS}, and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators.

math.AP↗
arXiv2016

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

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 $Lu_1=Lu_2=0$ in $Ω\cap B_1$, $u_1=u_2=0$ in $B_1\setminusΩ$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $Ω$. When $Ω$ is Lipschitz, we show that the quotient $u_1/u_2$ is Hölder continuous up to the boundary in $B_{1/2}$.

math.AP↗
arXiv2016

Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains

We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Δ)^su^m=0$ in $(0,\infty)\timesΩ$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminusΩ)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partialΩ$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,α}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $Ω$, both in bounded or unbounded domains.

math.AP↗
arXiv2016

Free boundary regularity in the parabolic fractional obstacle problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>\frac12$, the free boundary is a $C^{1,α}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)\in \partial\{u>φ\}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form \[u(x,t)-φ(x)=c_0\bigl((x-x_0)\cdot e+a(t-t_0)\bigr)_+^{1+s}+o\bigl(|x-x_0|^{1+s+α}+ |t-t_0|^{1+s+α}\bigr),\] with $c_0>0$, $e\in \mathbb{S}^{n-1}$, and $a>0$.

math.AP↗
arXiv2016

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

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 operator of order $2s$, with $s\in(0,1)$. First, in $C^{1,α}$ domains we show that all solutions $u$ are $C^s$ up to the boundary and that $u/d^s\in C^α(\barΩ)$, where $d$ is the distance to $\partialΩ$. In $C^1$ domains, solutions are in general not comparable to $d^s$, and we prove a boundary Harnack principle in such domains. Namely, we show that if $u_1$ and $u_2$ are positive solutions, then $u_1/u_2$ is bounded and Hölder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in non-divergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators \cite{CRS-obstacle}.

math.AP↗
arXiv2016

A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

math.AP↗
arXiv2016

Pohozaev identities for anisotropic integro-differential operators

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity $u/d^s|_{\partialΩ}$ plays the role that $\partial u/\partialν$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $Ω$, with $u=0$ in $\mathbb R^n\setminusΩ$, and $d$ is the distance to $\partialΩ$.

math.AP↗