arXiv subjects
Publications and source records attributed to Yannick Sire.
We show that bounded viscosity solutions of some nonlocal degenerate Isaacs type operators of order $2s$ are Hölder continuous, provided $s$ is sufficiently close to 1. As an application we obtain a Liouville theorem.
The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincaré-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincaré-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $ε$-regularity result for degenerate/singular elliptic operators on Poincaré-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincaré-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.
Suppose that $Σ^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $λ_1$ of the induced Laplace-Beltrami operator on $Σ$ satisfies $λ_1 \geq \frac{n}{2}+ a_n(Λ^6 + b_n)^{-1}$, where $a_n$ and $b_n$ are explicit dimensional constants and $Λ$ is an upper bound for the length of the second fundamental form of $Σ$. This provides the first explicitly computable improvement on Choi & Wang's lower bound $λ_1 \geq \frac{n}{2}$ without any further assumptions on $Σ$.
We consider a nematic liquid crystal flow with partially free boundary in a smooth bounded domain in $\mathbb{R}^2$. We prove regularity estimates and the global existence of weak solutions enjoying partial regularity properties, and a uniqueness result.
Let $(M,\textit{g},σ)$ be an $m$-dimensional closed spin manifold, with a fixed Riemannian metric $\textit{g}$ and a fixed spin structure $σ$; let $\mathbb{S}(M)$ be the spinor bundle over $M$. The spinorial Yamabe-type problems address the solvability of the following equation \[ D_{\textit{g}}ψ= f(x)|ψ|_{\textit{g}}^{\frac2{m-1}}ψ, \quad ψ:M\to\mathbb{S}(M), \ x\in M \] where $D_{\textit{g}}$ is the associated Dirac operator and $f:M\to\mathbb{R}$ is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when $m=2$, a solution corresponds to a conformal isometric immersion of the universal covering $\widetilde M$ into $\mathbb{R}^3$ with prescribed mean curvature $f$; meanwhile, for general dimensions and $f\equiv constant\neq0$, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold $(S^m,\textit{g})$ where the solution set of the spinorial Yamabe-type problem is not compact: $1).$ the geometric potential $f$ is constant (say $f\equiv1$) with the background metric $\textit{g}$ being a $C^k$ perturbation of the canonical round metric $\textit{g}_{S^m}$, which is not conformally flat somewhere on $S^m$; $2).$ $f$ is a perturbation from constant and is of class $C^2$, while the background metric $\textit{g}\equiv\textit{g}_{S^m}$.
We investigate a Sobolev map $f$ from a finite dimensional RCD space $(X, \dist_X, \meas_X)$ to a finite dimensional non-collapsed compact RCD space $(Y, \dist_Y, \mathcal{H}^N)$. If the image $f(X)$ is smooth in a weak sense (which is satisfied if $f_{\sharp}\meas_X$ is absolutely continuous with respect to the Hausdorff measure $\mathcal{H}^N$, or if $(Y, \dist_Y, \mathcal{H}^N)$ is smooth in a weak sense), then the pull-back $f^*g_Y$ of the Riemannian metric $g_Y$ of $(Y, \dist_Y, \mathcal{H}^N)$ is well-defined as an $L^1$-tensor on $X$, the minimal weak upper gradient $G_f$ of $f$ can be written by using $f^*g_Y$, and it coincides with the local slope for $\meas_X$-almost everywhere points in $X$ when $f$ is Lipschitz. In particular the last statement gives a nonlinear analogue of Cheeger's differentiability theorem for Lipschitz functions on metric measure spaces. Moreover these results allow us to define the energy of $f$. The energy coincides with the Korevaar-Schoen energy.In order to achieve this, we use a smoothing of $g_Y$ via the heat kernel embedding $Φ_t:Y \hookrightarrow L^2(Y, \mathcal{H}^N)$, which is established by Ambrosio-Portegies-Tewodrose and the first named author. Moreover we improve the regularity of $Φ_t$, which plays a key role. We show also that $(Y, \dist_Y)$ is isometric to the $N$-dimensional standard unit sphere in $\mathbb{R}^{N+1}$ and $f$ is a minimal isometric immersion if and only if $(X, \dist_X, \meas_X)$ is non-collapsed up to a multiplication of a constant to $\meas_X$, and $f$ is an eigenmap whose eigenvalues coincide with the essential dimension of $(X, \dist_X, \meas_X)$, which gives a positive answer to a remaining problem from a previous work by the first named author.
We derive several properties of the heat equation with the Hodge operator associated with the Rumin complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin's differential forms to construct a Calderón reproducing formula on Rumin forms.
We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $Ω$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial Ω)^α$, where $\partial Ω$ is the boundary of $Ω$ and $α\in (-1, \infty)$ is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.
We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $\mathbb{R}^d \times \mathbb{T}$, $d \geq 3$. This result can be regarded as the analogue of \cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in \cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.
We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. Despite its singularity we manage to invert the double layer potential restricted to the boundary thanks to a reflection technique for an approximate operator in $\mathbb{G}$. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.
In this paper we construct singular solutions to the critical Dirac equation on spheres. More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions constructed for the singular Yamabe problem in \cite{MP1 , Schoen1989}. Then we construct another kind of singular solutions admitting a great circle as a singular set. These solutions are the building blocks for singular solutions on a general Spin manifold.
We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the spectral projection operators on the boundary. Moreover, we derive lower bounds on the size of nodal sets for a variation of the Steklov spectral problem. We consider a generalized version of the Steklov problem by adding a non-smooth potential on the boundary but some of our results are new even without potential.
We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.
This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein inequalities for higher order Laplacians. We show also that the heat semi-group associated to powers larger than one of the laplacian does not satisfy the strict maximum principle in general. Finally, in a suitable range we provide several positive results.
We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in \cite{SSWZ} about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.
This note is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have been lately considered by Constantin and Ignatova in the framework of the SQG equation \cite{CI} in bounded domains and by two of the authors \cite{HungJuan} in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of Porous Medium Equations with pressure given by Riesz-type potentials .