arXiv subjects

Xavier Cabre

Publications and source records attributed to Xavier Cabre.

51 records · Page 3Linked to original sources

arXiv2012

Propagation in Fisher-KPP type equations with fractional diffusion in periodic media

We are interested in the time asymptotic location of the level sets of solutions to Fisher-KPP reaction-diffusion equations with fractional diffusion in periodic media. We show that the speed of propagation is exponential in time, with a precise exponent depending on a periodic principal eigenvalue, and that it does not depend on the space direction. This is in contrast with the Freidlin-Gärtner formula for the standard Laplacian.

math.AP↗

arXiv2012

Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation $(-Δ)^s u = f(u)$ in $\mathbb{R}^n$, for all fractions $0<s<1$ and all nonlinearities $f$. For every fractional power $s \in (0,1)$, we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension $n=3$ whenever $1/2 \leq s < 1$. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-Δu = f(u)$ in $\mathbb{R}^n$. It remains open for $n=3$ and $s<1/2$, and also for $n \geq 4$ and all $s$.

math.AP↗

arXiv2012

The influence of fractional diffusion in Fisher-KPP equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model.

math.AP↗

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 $5\leq n\leq 9$.

math.AP↗

arXiv2011

Geometric-type Sobolev inequalities and applications to the regularity of minimizers

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-Δu=g(u)$ in a bounded domain $Ω$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(Ω)$ bound for the extremal solution of $-Δu=λf(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\leq 4$, and that their boundedness is expected ---but still unkwown--- for $n\leq 9$.

math.AP↗

arXiv2011

Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions

This paper, which is the follow-up to part I, concerns the equation $(-Δ)^{s} v+G'(v)=0$ in $\mathbb{R}^{n}$, with $s \in (0,1)$, where $(-Δ)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a Lévy process. When $n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $\pm 1$ at $\pm \infty$) if and only if the potential $G$ has only two absolute minima in $[-1,1]$, located at $\pm 1$ and satisfying $G'(-1)=G'(1)=0$. Under the additional hypothesis $G"(-1)>0$ and $G"(1)>0$, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution. For $n\geq 1$, we prove some results related to the one-dimensional symmetry of certain solutions ---in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.

math.AP↗

arXiv2011

Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation

We establish the uniqueness of a saddle-shaped solution to the diffusion equation $-Δu = f(u)$ in all of $\mathbb{R}^{2m}$, where $f$ is of bistable type, in every even dimension $2m \geq 2$. In addition, we prove its stability whenever $2m \geq 14$. Saddle-shaped solutions are odd with respect to the Simons cone ${\mathcal C} = \{(x^1,x^2) \in \mathbb{R}^m \times \mathbb{R}^m : |x^1|=|x^2| \}$ and exist in all even dimensions. Their uniqueness was only known when $2m=2$. On the other hand, they are known to be unstable in dimensions 2, 4, and 6. Their stability in dimensions 8, 10, and 12 remains an open question. In addition, since the Simons cone minimizes area when $2m \geq 8$, saddle-shaped solutions are expected to be global minimizers when $2m \geq 8$, or at least in higher dimensions. This is a property stronger than stability which is not yet established in any dimension.

math.AP↗

arXiv2010

Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates

This is the first of two articles dealing with the equation $(-Δ)^{s} v= f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-Δ)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in $\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on $\partial \mathbb{R}^{n+1}_+=\mathbb{R}^{n}$. In this first article, we establish necessary conditions on the nonlinearity $f$ to admit certain type of solutions, with special interest in bounded increasing solutions in all of $\mathbb{R}$. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian ---in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

math.AP↗

arXiv2010

Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-Δ)^{1/2} u=f(u)$ in $\re^n$. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-Δu=f(u)$ in $\re^n$.

math.AP↗

arXiv2009

Regularity of minimizers of semilinear elliptic problems up to dimension four

We consider the class of semi-stable solutions to semilinear equations $-Δu=f(u)$ in a bounded smooth domain $Ω$ of $R^n$ (with $Ω$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n \leq 4$, we establish an priori $L^\infty$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $Ω$ is convex. This result was previously known only in dimensions $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n \leq 9$ the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case $Ω=B_R$ by a result of A. Capella and the author.

math.AP↗

arXiv2009

Qualitative properties of saddle-shaped solutions to bistable diffusion equations

We consider the elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in $\R^{2m}$. This is a solution which changes sign in $\R^{2m}$ and vanishes only on the Simons cone ${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m: |x^1|=|x^2|\}$. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when $2m=6$ every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.

math.AP↗

arXiv2009

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

math.AP↗

arXiv2009

Front propagation in Fisher-KPP equations with fractional diffusion

We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model.

math.AP↗

arXiv2008

Saddle-shaped solutions of bistable diffusion equations in all of $\mathbb{R}^{2m}$

We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that in dimension $2m=2$ there exists a saddle-shaped solution. This is a solution which changes sign in $\R^2$ and vanishes only on $\{|x_1|=|x_2|\}$. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension $2m=4$. More precisely, our main result establishes that if $2m=4$, every solution vanishing on the Simons cone $\{(x^1,x^2)\in\R^m\times\R^m : |x^1|=|x^2|\}$ is unstable outside of every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.

math.AP↗

arXiv2007

Regularity of radial minimizers of reaction equations involving the p-Laplacian

We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.

math.AP↗