arXiv subjects

Van Hoang Nguyen

Publications and source records attributed to Van Hoang Nguyen.

50 records · Page 3Linked to original sources

arXiv2017

An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in $\mathbf R^2$

Let $Ω$ be a smooth bounded domain in $\mathbf R^2$ and $λ^{\mathsf N} (Ω)$ the first non-zero Neumann eigenvalue of the operator $-Δ$ on $Ω$. In this paper, for any $γ\in [0, λ^{\mathsf N} (Ω) )$, we establish the following improved Moser-Trudinger inequality \[ \sup_{u} \int_Ω e^{2πu^2} dx < +\infty \] for arbitrary functions $u$ in $H^1(Ω)$ satisfying $\int_Ωu dx =0$ and $\|\nabla u\|_2^2 -α\|u\|_2^2 \leqslant 1$. Furthermore, this supremum is attained by some function $u^*\in H^1(Ω)$. This strengthens the results of Chang and Yang (J. Differential Geom. 27 (1988) 259-296) and of Lu and Yang (Nonlinear Anal. 70 (2009) 2992-3001).

math.AP↗

arXiv2017

The sharp Gagliardo--Nirenberg--Sobolev inequality in quantitative form

Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert, we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo--Nirenberg--Sobolev inequalities due to Del Pino and Dolbeault \cite{DelPino}.

math.FA↗

arXiv2017

A sharp Adams inequality in dimension four and its extremal functions

Let $Ω$ be a smooth oriented bounded domain in $\mathbb R^4$, $H_0^2(Ω)$ be the Sobolev space, and $λ_1(Ω)= \inf \{\|Δu\|_2^2 : u\in H_0^2(Ω), \|u\|_2 =1\}$ be the first eigenvalue of the bi-Laplacian operator $Δ^2$ on $Ω$. For $α\in [0,λ_1(Ω))$, we define $\|u\|_{2,α}^2 = \|Δu\|_2^2 - α\|u\|_2^2$, for $u \in H_0^2(Ω)$. In this paper, we will prove the following inequality \[ \sup_{u\in H_0^2(Ω),\, \|u\|_{2,α} \leq 1} \int_Ω e^{32 π^2 u(x)^2} dx < \infty. \] This strengthens a recent result of Lu and Yang \cite{LuYang}. We also show that there exists a function $u^*\in H_0^2(Ω)\cap C^4(\overlineΩ)$ such that $\|u^*\|_{2,α} =1$ and the supremum above is attained by $u^*$. Our proofs are based on the blow-up analysis method.

math.FA↗

arXiv2016

New approach to the affine Pólya-Szegö principle and the stability version of the affine Sobolev inequality

Inspired by a recent work of Haddad, Jiménez and Montenegro, we give a new and simple approach to the recently established general affine Pólya-Szegö principle. Our approach is based on the general $L_p$ Busemann-Petty centroid inequality and does not rely on the general $L_p$ Petty projection inequality or the solution of the $L_p$ Minkowski problem. A Brothers-Ziemer-type result for the general affine Pólya-Szegö principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic--Sobolev inequality.

math.FA↗

arXiv2016

Sharp reversed Hardy-Littlewood-Sobolev inequality on $\mathbb R^n$

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] for any nonnegative functions $f\in L^p(\mathbb R^n)$, $g\in L^r(\mathbb R^n)$, and $p,r\in (0,1)$, $λ> 0$ such that $1/p + 1/r -λ/n =2$. We will also explore some estimates for $\mathscr C_{n,p,r}$ and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.

math.AP↗

arXiv2016

An improvement for the sharp Adams inequalities in bounded domains and whole space $\mathbb{R}^n$

We prove an improvement for the sharp Adams inequality in $W^{m,\frac nm}_0(Ω)$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our method gives an alternative approach to a Concentration--Compactness principle in $W^{m,\frac nm}_0(Ω)$ recently established by do Ó and Macedo. Moreover, when $m$ is odd, we obtain an improvement for their result by finding the best exponent in this principle. Our approach also is successfully applied to whole space $\mathbb{R}^n$ to establish an improvement for the sharp Adams inequalities in $W^{m,\frac nm}(\mathbb{R}^n)$ due to Ruf, Sani, Lam, Lu, Fontana and Morpurgo. This type of improvement is still unknown, in general, except the special case $m=1$ due to do Ó, de Souza, de Medeiros and Severo. Our method is a further development for the method of $\check{\rm C}$erny, Cianchi and Hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives.

math.AP↗

arXiv2016

Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$ \[ \int_{\mathbb R_+^n} \int_{\partial \mathbb R_+^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\partial \mathbb R_+^n)} \, \|g\|_{L^r(\mathbb R_+^n)} \] for any nonnegative functions $f\in L^p(\partial \mathbb R_+^n)$, $g\in L^r(\mathbb R_+^n)$, and $p,r\in (0,1)$, $λ> 0$ such that $(1-1/n)1/p + 1/r -(λ-1) /n =2$. Some estimates for $\mathscr C_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.

math.AP↗

arXiv2016

Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities

We prove a trace Hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of Alvino et al. and of Tzirakis on the upper half space. We also prove some trace Hardy-Sobolev-Maz'ya type inequalities which generalize the recent results of Filippas et al.. In applications, we derive some Hardy type inequalities and Hardy-Sobolev-Maz'ya type inequalities for fractional Laplacian. Finally, we prove the logarithmic Sobolev trace inequalities and logarithmic Hardy trace inequalities on the upper half spaces. The best constants in these inequalities are explicitly computed in the radial case.

math.FA↗

arXiv2015

Improved $L_p-$mixed volume inequality for convex bodies

A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csiszár-Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the $L_p-$Brunn-Minkowski inequality is also proved as a corollary.

math.FA↗

arXiv2015

Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half space via mass transport and consequences

By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$ equipped with the weight $ω(x) = x_n^a$, $a\geq 0$. It amounts to work with the fractional dimension $n_a = n+a$. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domain $\mathbf{R}^{n-m}\times \mathbf{R}^m_+$ and the weights are $ω(x,t) = t_1^{a_1}\dots t_m^{a_m}$, where $n\geq m$, $m\geq 0$ and $a_1,\cdots,a_m\geq 0$. A weighted $L^p$-logarithmic Sobolev inequality is derived from these inequalities.

math.FA↗

arXiv2014

Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities

This work focuses on an improved fractional Sobolev inequality with a remainder term involving the Hardy-Littlewood-Sobolev inequality which has been proved recently. By extending a recent result on the standard Laplacian to the fractional case, we offer a new, simpler proof and provide new estimates on the best constant involved. Using endpoint differentiation, we also obtain an improved version of a Moser-Trudinger-Onofri type inequality on the sphere. As an immediate consequence, we derive an improved version of the Onofri inequality on the Euclidean space using the stereographic projection.

math.FA↗

arXiv2014

A local proof of the dimensional Prékopa's theorem

The aim of this paper is to find an expression for second derivative of the function $ϕ(t)$ defined by $$ϕ(t) = \lt(\int_V \vphi(t,x)^{-β} dx\rt)^{-\frac1{\be -n}},\qquad β\not= n,$$ where $U\subset \R$ and $V\subset \R^n$ are open bounded subsets, and $\vphi: U\times V\to \R_+$ is a $C^2-$smooth function. As a consequence, this result gives us a direct proof of the dimensional Prékopa's theorem based on a local approach.

math.FA↗

arXiv2013

Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem

We give a new approach, inspired by Hörmander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the Brunn-Minkowski inequalities. We also present several applications of these variance inequalities, including reverse Hölder inequalities for convex functions, weighted Brascamp-Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.

math.FA↗

arXiv2013

Entropy jumps for isotropic log-concave random vectors and spectral gap

We prove a quantitative dimension-free bound in the Shannon-Stam Entropy inequality for the convolution of two log-concave distributions in dimension d interms of the spectral gap of the density. The method relies on the analysis of the Fisher Information production, which is the second derivative of the Entropy along the (normalized) Heat semi-group. We also discuss consequences of our result in the study of the isotropic constant of log-concave distributions (slicing problem).

math.FA↗