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Xavier Ros-Oton

Publications and source records attributed to Xavier Ros-Oton.

At least 73 records · Page 4Linked to original sources
arXiv2015

The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $Ω$, $u=0$ in $\R^n\setminusΩ$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,dρ\int_{S^{n-1}}\,da(ω)\, \frac{ 2u(x)-u(x+ρω)-u(x-ρω)}{ρ^{1+2s}}.$$ Here, $a$ is any measure on~$S^{n-1}$ (a prototype example for~$L$ is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When $a\in C^\infty(S^{n-1})$ and $g$ is $C^\infty(Ω)$, solutions are known to be $C^\infty$ inside~$Ω$ (but not up to the boundary). However, when $a$ is a general measure, or even when $a$ is $L^\infty(S^{n-1})$, solutions are only known to be $C^{3s}$ inside $Ω$. We prove here that, for general measures $a$, solutions are $C^{1+3s-ε}$ inside $Ω$ for all $ε>0$ whenever $Ω$ is convex. When $a\in L^{\infty}(S^{n-1})$, we show that the same holds in all $C^{1,1}$ domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample for which $u$ is \emph{not} $C^{3s+ε}$ for any $ε>0$ -- even if $g$ and $Ω$ are $C^\infty$.

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arXiv2015

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable Lévy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $Ω$, $u=0$ in $\mathbb R^n\setminusΩ$, satisfy $u/d^s\in C^{s+γ}(\overlineΩ)$, where $d$ is the distance to $\partialΩ$ and $f\in C^γ$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.

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arXiv2015

The Pohozaev identity for the fractional Laplacian

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-Δ)^s u = f(u)$ in $Ω$, $u \equiv 0$ in $\mathbb R^n\setminusΩ$. 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 $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$, where $δ(x)={\rm dist}(x,\partialΩ)$. In the fractional Pohozaev identity, the function $u/δ^s|_{\partialΩ}$ plays the role that $\partial u/\partialν$ plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over $\partialΩ$) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.

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arXiv2015

Local integration by parts and Pohozaev identities for higher order fractional Laplacians

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-Δ)^s$ 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 $(-Δ)^sϕ=λϕ$ in $Ω$, $ϕ\equiv0$ in $\mathbb R^n\setminusΩ$.

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arXiv2015

Entire solutions to semilinear nonlocal equations in $\RR^2$

We consider entire solutions to $L u= f(u)$ in $\RR^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.

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arXiv2015

Sobolev and isoperimetric inequalities with monomial weights

We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ is a real number for each $i=1,...,n$, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure $dx$ replaced by $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, these inequalities are exactly the classical ones in the Euclidean space $\mathbb R^D$ (with no weight) when written for axially symmetric functions and domains in $\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$.

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arXiv2015

Nonlocal elliptic equations in bounded domains: a survey

In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }Ω\\ u&=&g&\textrm{in }\mathbb R^n\backslashΩ\end{array}\right.\] for nonlocal operators of the form \[Lu(x)=\textrm{PV}\int_{\mathbb R^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)dy.\] We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators $L$ in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field.

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arXiv2014

Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric Lévy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^α$ then $u$ belong to $C^{α+2s}$ whenever $α+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-ε}$ for all $ε>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $\mathbb R^n\setminusΩ$, in $C^{1,1}$ domains $Ω$. We show that solutions $u$ satisfy $u/d^s\in C^{s-ε}(\overlineΩ)$ for all $ε>0$, where $d$ is the distance to $\partialΩ$. Finally, we show that our results are sharp by constructing two counterexamples.

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arXiv2014

Boundary regularity for the fractional heat equation

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $C^{1,1}$ domains. More precisely, we consider solutions to $\partial_t u + (-Δ)^s u=0 \textrm{ in }Ω,\ t > 0$, with zero Dirichlet conditions in $\mathbb{R}^n\setminus Ω$ and with initial data $u_0\in L^2(Ω)$. Using the results of the second author and Serra for the elliptic problem, we show that for all $t>0$ we have $u(\cdot, t)\in C^s(\mathbb{R}^n)$ and $u(\cdot, t)/δ^s \in C^{s-ε}(\overlineΩ)$ for any $ε> 0$ and $δ(x) = \textrm{dist}(x,\partialΩ)$. Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable Lévy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.

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arXiv2014

Nonlocal problems with Neumann boundary conditions

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside $Ω$, decreasing energy, and convergence to a constant as $t\to \infty$. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition~$\partial_νu=0$ on~$\partialΩ$ consists in the nonlocal prescription $$ \int_Ω\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} x\in\R^n\setminus\overlineΩ.$$

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arXiv2014

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem $(-Δ)^su=λe^u$ in a bounded domain $Ω\subset\mathbb R^n$, where $λ$ is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions $n\leq7$ for all $s\in(0,1)$ whenever $Ω$ is, for every $i=1,...,n$, convex in the $x_i$-direction and symmetric with respect to $\{x_i=0\}$. The same holds if $n=8$ and $s\gtrsim0'28206...$, or if $n=9$ and $s\gtrsim0'63237...$. These results are new even in the unit ball $Ω=B_1$.

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arXiv2013

Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] 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 $σ\in(0,2)$ in case $a_{ij}\equiv0$ 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 critical and supercritical nonlinearities (where the criticality condition is in relation to $n$ and $σ$). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian $(-Δ)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.

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arXiv2013

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem $(-Δ)^s u=λf(u)$ in $Ω$, $u\equiv0$ in $\R^n\setminusΩ$, where $λ>0$ 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 $n<10s$. In the limit $s\uparrow1$, $n<10$ is optimal. In addition, we show that the extremal solution is $H^s(\R^n)$ in any dimension whenever the domain is convex. To obtain some of these results we need $L^q$ estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $L^p$ data. We prove optimal $L^q$ and $C^β$ estimates, depending on the value of $p$. These estimates follow from classical embedding results for the Riesz potential in $\R^n$. Finally, to prove the $H^s$ regularity of the extremal solution we need an $L^\infty$ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.

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arXiv2013

Sharp isoperimetric inequalities via the ABP method

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.

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arXiv2012

Euclidean balls solve some isoperimetric problems with nonradial weights

In this note we present the solution of some isoperimetric problems in open convex cones of $\R^n$ 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.

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arXiv2012

The Dirichlet problem for the fractional Laplacian: regularity up to the boundary

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 $(-Δ)^s u = g$ in $Ω$, $u \equiv 0$ in $\R^n\setminusΩ$, for some $s\in(0,1)$ and $g \in L^\infty(Ω)$, then $u$ is $C^s(\R^n)$ and $u/δ^s|_Ω$ is $C^α$ up to the boundary $\partialΩ$ for some $α\in(0,1)$, where $δ(x)={\rm dist}(x,\partialΩ)$. 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 $u/δ^s$. Namely, the $C^β$ norms of $u$ and $u/δ^s$ in the sets $\{x\inΩ: δ(x)\geqρ\}$ are controlled by $Cρ^{s-β}$ and $Cρ^{α-β}$, respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian \cite{RS-CRAS,RS}.

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arXiv2012

Regularity of stable solutions up to dimension 7 in domains of double revolution

We consider the class of semi-stable positive solutions to semilinear equations $-Δu=f(u)$ in a bounded domain $Ω\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and of the last $n-m$ variables. We assume $2\leq m\leq n-2$. When the domain is convex, we establish a priori $L^p$ and $H^1_0$ bounds for each dimension $n$, with $p=\infty$ when $n\leq7$. These estimates lead to the boundedness of the extremal solution of $-Δu=λf(u)$ in every convex domain of double revolution when $n\leq7$. The boundedness of extremal solutions is known when $n\leq3$ for any domain $Ω$, in dimension $n=4$ when the domain is convex, and in dimensions $5\leq n\leq9$ in the radial case. Except for the radial case, our result is the first partial answer valid for all nonlinearities $f$ in dimensions $5\leq n\leq 9$.

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