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Yannick Sire

Publications and source records attributed to Yannick Sire.

At least 55 records · Page 3Linked to original sources
arXiv2021

Multiplier theorems via martingale transforms

We develop a new approach to prove multiplier theorems in various geometric settings. The main idea is to use martingale transforms and a Gundy-Varopoulos representation for multipliers defined via a suitable extension procedure. Along the way, we provide a probabilistic proof of a generalization of a result by Stinga and Torrea, which is of independent interest. Our methods here also recover the sharp $L^p$ bounds for second order Riesz transforms by a liming argument.

math.PR↗
arXiv2021

Uniform Sobolev Estimates on compact manifolds involving singular potentials

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author \cite{KRS} for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo \cite{DKS} for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the the first, third and fourth authors \cite{BSS} , Hassell and Tacy \cite{HassellTacy}, the first and fourth author \cite{SBLog}, and Hickman \cite{Hickman} as well as analogues of the improved uniform Sobolev estimates of \cite{BSSY} and \cite{Hickman} involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author \cite{SHSo}. For general Riemannian manifolds we improve the earlier results in \cite{BSS} by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.

math.AP↗
arXiv2020

Singular solutions for the constant $Q$-curvature problem

This paper is devoted to the construction of weak solutions to the singular constant $Q$-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of sufficiently large dimension with constant (positive) $Q$-curvature.

math.AP↗
arXiv2020

Powers Of Generators On Dirichlet Spaces And Applications To Harnack Principles

We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific geometries; a second one is to provide new results on rather general metric measured spaces that were not considered before and fall naturally in the theory of Dirichlet spaces. The main tool is using the approach based on subordination and semi-groups by Stinga and Torrea. Assuming more on the Dirichlet space, we derive several applications to PDEs such as Harnack and Boundary Harnack principles.

math.AP↗
arXiv2020

Classification of Blow-ups and Monotonicity Formula for Half Laplacian Nonlinear Heat Equation

We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ and $ 1<p<p_{*} (n)$ where $ p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension to the half-space.

math.AP↗
arXiv2020

Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions

For a smooth bounded domain $Ω\subseteq\mathbb{R}^n$, $n\geq 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\partial w}{\partialτ} =Δw^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot, τ) = 0$ on $\partialΩ$. Using the parabolic gluing method, we prove existence of an initial data $w_0$ such that the corresponding solution has extinction rate of the form $$\|w(\cdot, τ)\|_{L^\infty(Ω)} = γ_0(T-τ)^{\frac{n+2}{4}}\left|\ln(T-τ)\right|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ as $t\to T^-$, here $T > 0$ is the finite extinction time of $w(x, τ)$. This generalizes and provides rigorous proof of a result of Galaktionov and King \cite{galaktionov2001fast} for the radially symmetric case $Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.

math.AP↗
arXiv2020

Singular Yamabe metrics by equivariant reduction

We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients.

math.AP↗
arXiv2020

Regularity for parabolic systems with critical growth in the gradient and applications

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.

math.AP↗
arXiv2020

A variational analysis of the spinorial Yamabe equation on product manifolds

This work is devoted to the analysis of the Yamabe problem on Spin manifolds and some applications to CMC immersions. Despite the efforts of many authors, very little is known on the existence of Yamabe metrics on general Spin manifolds. Motivated to bubbling phenomena for the Riemannian problem and recent multiplicity results in this setting, we investigate special spinorial Yamabe metrics on product manifolds developing a bubbling analysis which has independent interest in the present setting.

math.AP↗
arXiv2020

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)\; \quad\textrm{or} \ \textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y))$$ as $\varepsilon\to 0$. Finally, we derive $C^{0,α}$ and $C^{1,α}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

math.AP↗
arXiv2020

Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. $$ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y)) $$ as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

math.AP↗
arXiv2020

$L^p$ eigenfunction bounds for fractional Schrödinger operators on manifolds

This paper is dedicated to $L^p$ bounds on eigenfunctions of a Schödinger-type operator $(-Δ_g)^{α/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-Δ_g)^{α/2}$ is defined spectrally in terms of the eigenfunctions of $-Δ_g$. We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation $(\partial_t^2+(-Δ_g)^{α/2}+V)u=0$. The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao and Shao-Yao play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the "uniform Sobolev estimates" by Kenig-Ruiz-Sogge. These enable us to handle the critically singular potentials $V$ and prove the quasimode estimates.

math.AP↗
arXiv2020

Free boundary problems involving singular weights

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+γ}$ regularity result for solutions at their singular free boundary points. We also show a corresponding non-degeneracy estimate.

math.AP↗
arXiv2019

Uniformization Theorems: Between Yamabe and Paneitz

This paper is devoted to several existence results for a generalized version of the Yamabe problem. First, we prove the remaining global cases for the range of powers $γ\in (0,1)$ for the generalized Yamabe problem introduced by Gonzalez and Qing. Second, building on a new approach by Case and Chang for this problem, we prove that this Yamabe problem is solvable in the Poincaré-Einstein case for $γ\in (1,\min\{2,n/2\})$ provided the associated fractional GJMS operator satisfies the strong maximum principle.

math.AP↗
arXiv2019

Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from \cite{CH2} and arguments borrowed from \cite{HZ, Zhang}. As an application, we prove the small data global existence for any power $p\in(1, 1+\frac{4}{n-1})$ for the shifted wave equation in this setting, involving nonlinearities of the form $\pm|u|^p$ or $\pm|u|^{p-1}u$, which answers partially an open question raised in \cite{SSW}.

math.AP↗
arXiv2019

The Strauss conjecture on negatively curved backgrounds

This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $p\in (1, 1+\frac{4}{n-1}]$ for the shifted wave equation on hyperbolic space ${\mathbb H}^n$ involving nonlinearities of the form $\pm |u|^p$ or $\pm|u|^{p-1}u$. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for the constant curvature case of Anker-Pierfelice and Metcalfe-Taylor. We also discuss the role of curvature and state a couple of open problems. Finally, in an appendix, we give an alternate proof of dispersive estimates of Tataru for ${\mathbb H}^3$ and settle a dispute, in his favor, raised in Metcalfe-Taylor about his proof. Our proof is slightly more self-contained than the one in Tataru since it does not make use of heavy spherical analysis on hyperbolic space such as the Harish-Chandra $c$-function; instead it relies only on simple facts about Bessel potentials.

math.AP↗