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Yannick Sire

Publications and source records attributed to Yannick Sire.

At least 91 records · Page 5Linked to original sources
arXiv2016

Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Ω\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Δ)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the singular limit $\varepsilon\to 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Ω$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^\prime$-perimeter in $Ω$ for every $s^\prime\in(0,1/2)$.

math.AP↗
arXiv2016

On some pointwise inequalities involving nonlocal operators

The purpose of this paper is three-fold: first, we survey on several known pointwise identities involving fractional operators; second, we propose a unified way to deal with those identities; third, we prove some new pointwise identities in different frameworks in particular geometric and infinite-dimensional ones.

math.AP↗
arXiv2016

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-Δ)^s u=0$, $0<s<1$, posed in the whole space $\mathbb{R}^N$ with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions, which is given in one direction by the representation formula, in the other one by the initial trace. We review many of the typical properties of the solutions, in particular we prove optimal pointwise estimates and new Harnack inequalities.

math.AP↗
arXiv2016

An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an \emph{a-posteriori} format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functional equation for the parameterization of the torus satisfying the invariance equations and for parametrization of directions invariant under the linearizatation. The iterative method does not use transformation theory or action-angle variables. It does not assume that the system is close to integrable. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples, including the scalar Boussinesq equation and the Boussinesq system so that we construct {\sl small amplitude} tori for the equations, which are even in the spatial variable. Note that the equations we use as examples are ill-posed.

math.DS↗
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↗
arXiv2015

Symmetry results for fractional elliptic systems and related problems

We study elliptic gradient systems with fractional laplacian operators on the whole space $$ (- Δ)^\mathbf s \mathbf u =\nabla H (\mathbf u) \ \ \text{in}\ \ \mathbf{R}^n,$$ where $\mathbf u:\mathbf{R}^n\to \mathbf{R}^m$, $H\in C^{2,γ}(\mathbf{R}^m)$ for $γ> \max(0,1-2\min \left \{s_i \right \})$, $\mathbf s=(s_1,\cdots,s_m)$ for $0<s_i<1$ and $\nabla H (\mathbf u)=(H_{u_i}(u_1, u_2,\cdots,u_m))_{i}$. We prove De Giorgi type results for this system for certain values of $\mathbf s$ and in lower dimensions, i.e. $n=2,3$. Just like the local case, the concepts of orientable systems and $H-$monotone solutions, established in [18], play the key role in proving symmetry results. In addition, we provide optimal energy estimates, a monotonicity formula, a Hamiltonian identity and various Liouville theorems.

math.AP↗
arXiv2015

A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation $$i\partial_tu-(-Δ)^σu+|u|^{2}u=0,$$ where $σ\in (\frac12,1)$ and the operator $(-Δ)^σ$ is the fractional Laplacian of symbol $|ξ|^{2σ}$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k|^{2σ}-ω^{2σ})}Q_{ω,k}(x-2tσ|k|^{2σ-2}k),\quad k\in\mathbb{R},\ ω>0$$ by a rather involved variational argument.

math.AP↗
arXiv2015

Singular Perturbation Problem in Boundary/Fractional Combustion

Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform Hölder regularity is shown, we pass to the limit to study the free boundary problem it leads to.

math.AP↗
arXiv2015

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $Ω$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $Ω$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.

math.AP↗
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.

math.AP↗
arXiv2015

On Fractional Schrodinger Equations in sobolev spaces

Let $σ\in(0,1)$ with $σ\neq\frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: $$i\partial_tu+(-Δ)^σu+μ|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-Δ)^σ$ is the Fourier multiplier of symbol $|ξ|^{2σ}$, and $μ=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

math.AP↗
arXiv2014

Bubbling solutions for nonlocal elliptic problems

We investigate bubbling solutions for the nonlocal equation \[ A_Ω^s u =u^p,\ u >0 \quad \mbox{in } Ω, \] under homogeneous Dirichlet conditions, where $Ω$ is a bounded and smooth domain. The operator $A_Ω^s$ stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases $s \in (0,1)$ and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial Ω$ and for the restricted fractional Laplacian, we prescribe $u=0$ on $\mathbb R^n \setminus Ω$. We construct solutions when the exponent $p = (n+2s)/(n-2s) \pm \varepsilon$ is close to the critical one, concentrating as $\varepsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain

math.AP↗
arXiv2014

Conformal Spectrum and Harmonic maps

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing $λ_1$ into the sphere is harmonic, establishing a link between conformal spectrum and harmonic maps.

math.DG↗
arXiv2014

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

math.AP↗