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Emmanuel Russ

Publications and source records attributed to Emmanuel Russ.

At least 19 recordsLinked to original sources
arXiv2025

Infinite graphs satisfying the Bakry-Emery curvature condition CD(0, n): The modified heat equation and applications to geometric analysis

Let G = (V, p, $\mu$) be a (finite or infinite) weighted graph with bounded geometry. Assuming that G satisfies the classical curvaturedimension condition of Bakry-Emery CD(K, n) with K $\ge$ 0 (for the usual Laplacian), we prove that the doubling volume property holds. One of the key points is to establish the existence and uniqueness of solutions of a modified non linear heat equation which replaces the standard one usually used in the case of Riemannian manifolds. Li-Yau and Harnack estimates for the solutions of this modified heat equation are obtained. We also provide explicit examples of Cayley graphs satisfying our assumptions.

math.DG↗
arXiv2024

A Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition

We prove a Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition, provided that the $\beta$ parameter in the Robin condition is large enough. The proof relies on a compactness argument, on the convergence of Robin eigenvalues to Dirichlet eigenvalues when $\beta$ goes to infinity, and on a strict Faber-Krahn inequality under Dirichlet boundary condition. We also show the existence and uniqueness of drifts $v$ satisfying some $L^\infty$ constraints and minimizing or maximizing the principal eigenvalue of $-\Delta+v\cdot\nabla$ in a fixed domain and with a fixed parameter $\beta>0$ in the Robin condition.

math.AP↗
arXiv2024

Reverse inequalities for quasi-Riesz transform on the Vicsek cable system

This work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ is false for all $p\in [1,2)$. Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form $\left\Vert \Delta^{\gamma}e^{-\Delta}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$, in the (unbounded) Vicsek cable system, for $p\in (1,+\infty)$ and $\gamma>0$. These reverse inequalities are strongly related to the problem of $L^p$ boundedness of the operators $\nabla e^{-\Delta}\Delta^{-\varepsilon}$, the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of $\gamma\in (0,1)$ and $p\in (1,+\infty)$ such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these.

math.AP↗
arXiv2022

Reverse inequality for the riesz transforms on Riemannian manifolds

Let $M$ be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and $L^q$ scaled Poincar\'e inequalities on suitable remote balls for some $q<2$. We prove the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all $p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when $M$ has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest. The second part of this work deals with analogous questions in fractal-like cable systems. In this framework, it was already proved by Chen, Coulhon, Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ may be false for all $p\in [1,2)$. Following a recent joint work by the two authors and Yang, we examine the validity of inequalities of the form $\left\Vert \Delta^{\gamma}e^{-\Delta}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$. In the Vicsek case, we give the optimal range of $p$ for which this inequality holds.

math.AP↗
arXiv2020

Solvability In Weighted Lebesgue Spaces of the Divergence Equation with Measure Data

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, a positive Radon measure $\mu$ 0 in $\Omega$ and a (signed) Radon measure $\mu$ on $\Omega$ satisfying $\mu$($\Omega$) = 0 and |$\mu$| $\kappa$$\mu$ 0 , the possibility of solving the equation div u = $\mu$ by a vector field u satisfying |u| $\kappa$w on $\Omega$ (where w is an integrable weight only related to the geometry of $\Omega$ and to $\mu$ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with measure data, and also construct a weight w that relies in a softer way on the geometry of $\Omega$, improving its behavior (and hence the a priori behavior of the solution we construct) substantially in some instances. The method used in this paper follows a constructive approach of Bogovskii type.

math.AP↗
arXiv2019

Hardy spaces on Riemannian manifolds with quadratic curvature decay

Let (M, g) be a complete Riemannian manifold. Assume that the Ricci curvature of M has quadratic decay and that the volume growth is strictly faster than quadratic. We establish that the Hardy spaces of exact 1-differential forms on M , introduced in [4], coincide with the closure in L p of R(d) $\cap$ L p ($\Lambda$ 1 T * M) when 1 < p < $\nu$, where $\nu$ > 2 is related to the volume growth. The range of p is optimal. This result applies, in particular, when M has a finite number of Euclidean ends.

math.CA↗
arXiv2018

Existence and Regularity of Optimal Shapes for Elliptic Operators with Drift

This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = --$\Delta$+V (x)\cdot \nabla with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue $\lambda$\_1($\Omega$, V) for a bounded quasi-open set $\Omega$ which enjoys similar properties to the case of open sets. Then, given m > 0 and $\tau$ $\ge$ 0, we show that the minimum of the following non-variational problem min $\lambda$\_1($\Omega$, V) : $\Omega$ $\subset$ D quasi-open, |$\Omega$| $\le$ m, |V|\_{\infty} $\le$ $\tau$. is achieved, where the box D $\subset$ R^d is a bounded open set. The existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape $\Omega$ * solving the minimization problem min $\lambda$\_1($\Omega$, $\Phi$) : $\Omega$ $\subset$ D quasi-open, |$\Omega$| $\le$ m , where $\Phi$ is a given Lipschitz function on D. We prove that the topological boundary $\partial$$\Omega$ * is composed of a regular part which is locally the graph of a C ^{1,$\alpha$} function and a singular part which is empty if d < d * , discrete if d = d * and of locally finite H^{d--d *} Hausdorff measure if d > d * , where d * $\in$ {5, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x $\in$ $\partial$$\Omega$ * $\cap$ $\partial$D, $\partial$$\Omega$ * is C^{ 1,$\alpha$} in a neighborhood of x, for some $\alpha$ $\le$ 1 /2. This last result is optimal in the sense that C ^{1,1/2} is the best regularity that one can expect.

math.AP↗
arXiv2017

Optimization of some eigenvalue problems with large drift

This paper is concerned with eigenvalue problems for non-symmetric elliptic operators with large drifts in bounded domains under Dirichlet boundary conditions. We consider the minimal principal eigenvalue and the related principal eigenfunction in the class of drifts having a given, but large, pointwise upper bound. We show that, in the asymptotic limit of large drifts, the maximal points of the optimal principal eigenfunctions converge to the set of points maximizing the distance to the boundary of the domain. We also show the uniform asymptotic profile of these principal eigenfunctions and the direction of their gradients in neighborhoods of the boundary.

math.AP↗
arXiv2017

Approximation in higher-order Sobolev spaces and Hodge systems

Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $\varphi$ be a differential $l$-form on ${\mathbb R}^d$ with $\dot{W}^{1,d}$ coefficients. It was proved by Bourgain and Brezis (\cite[Theorem 5]{MR2293957}) that there exists a differential $l$-form $\psi$ on ${\mathbb R}^d$ with coefficients in $L^{\infty}\cap \dot{W}^{1,d}$ such that $d\varphi=d\psi$. Bourgain and Brezis also asked whether this result can be extended to differential forms with coefficients in the fractional Sobolev space $\dot{W}^{s,p}$ with $sp=d$. We give a positive answer to this question, in the more general context of Triebel-Lizorkin spaces, provided that $d-\kappa\leq l\leq d-1$, where $\kappa$ is the largest positive integer such that $\kappa<\min(p,d)$. The proof relies on an approximation result for functions in $\dot{W}^{s,p}$ by functions in $\dot{W}^{s,p}\cap L^{\infty}$, even though $\dot{W}^{s,p}$ does not embed into $L^{\infty}$ in this critical case.

math.CA↗
arXiv2017

Lifting in Besov Spaces

Let $\Omega$ be a smooth bounded domain in $\mathbb R^n$ and u be a measurable function on $\Omega$ such that $|u(x)|=1$ almost everywhere in $\Omega$. Assume that u belongs to the $B^s_{p,q}(\Omega)$ Besov space. We investigate whether there exists a real-valued function $\varphi \in B^s_{p,q}$ such that $u=e^{i\varphi}$. This extends the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The analysis of this lifting problem leads us to prove some interesting new properties of Besov spaces, in particular a non restriction property when $q>p$.

math.CA↗
arXiv2015

Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound

We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on $L^p$ for $1 \textless{} p \textless{} 2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this.In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1 \textless{} p \textless{} 2$. This yields a full picture of the ranges of $p\in (1,+\infty)$ for which respectively the Riesz transform is $L^p$ -bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

math.CA↗
arXiv2015

Removable singularities for div v = f in weighted Lebesgue spaces

Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}\_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\_{1/w}$, $1\textless{}p\textless{}+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.

math.CA↗
arXiv2014

Comparison results for semilinear elliptic equations using a new symmetrization method

In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain $Ω$ of $\R^n$ and the solutions of some radially symmetric equations in the equimeasurable ball $Ω^*$. The coefficients of the symmetrized equations in~$Ω^*$ satisfy similar constraints as the original ones in~$Ω$. We consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient. Lastly, we show some improved quantified comparisons when the original domain is not a ball. The method is based on a symmetrization of the second-order terms.

math.AP↗
arXiv2013

Composition operators on generalized Hardy spaces

Let $Ω_1,Ω_2\subset {\mathbb C}$ be bounded domains. Let $ϕ:Ω_1\rightarrow Ω_2$ holomorphic in $Ω_1$ and belonging to $W^{1,\infty}_{Ω_2}(Ω_1)$. We study the composition operators $f\mapsto f\circϕ$ on generalized Hardy spaces on $Ω_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $ϕ$, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact. Some of our results are new even for Hardy spaces of analytic functions.

math.FA↗
arXiv2012

Sobolev and Hardy-Sobolev spaces on graphs

Let $Γ$ be a graph. Under suitable geometric assumptions on $Γ$, we give several equivalent characterizations of Sobolev and Hardy-Sobolev spaces on $Γ$, in terms of maximal functionals, Hajł asz type functionals or atomic decompositions. As an application, we study the boundedness of Riesz transforms on Hardy spaces on $Γ$. This gives the discrete counterpart of the corresponding results on Riemannian manifolds.

math.CA↗
arXiv2011

Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat semigroup satisfying off-diagonal decays. We don't require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another one using functionals). We also give a chain rule and study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non bounded geometry) and even in the Euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

math.CA↗