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Van Hoang Nguyen

Publications and source records attributed to Van Hoang Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities

In 2018, V. Maz'ya proposed 75 open problems in analysis and PDEs. One of them is about the best constant for the Heisenberg Uncertainty Principle for solenoidal vector fields motivated by questions in hydrodynamics. This was answered by Cazacu, Flynn and Lam in dimension two and subsequently by Hamamoto in all dimensions $N\ge3$ by establishing sharp inequalities with explicit constants. Nevertheless, the much harder stability problem remains open in all dimensions. The first main result in this paper is to establish a sharp stability hierarchy by proving that its deficit controls the distance to the set of extremals, with the sharp stability constant $\frac12(N-\sqrt{N^2-4N+12})$ for $N\ge4$ and $1$ for $N=3$, together with chains of remainder terms measuring the distance to explicit larger families of poloidal and toroidal fields. The second main result is the second order $L^2$-Caffarelli--Kohn--Nirenberg inequality with the weights $|x|^{-2a}$ and $|x|^{-2b}$ on the line $1+a+b=0$, for which we obtain the stability constant $\frac12\min\{4(1+a),\sqrt{(N+2a)^2+4N-4}-(N+2a)\}$. This CKN inequality is sharp whenever the second number is the smaller one. We develop a novel method: the fourth order one-dimensional problems attached to the spherical modes are transformed, by a Fourier--Hankel transform of real order, into first order inequalities whose sharp stability is that of weighted Gaussian Poincaré inequalities. The inverse transform produces the extremals in terms of Kummer's function. As a byproduct of our new approach, we also obtain substantially simpler proofs, with sharp remainder terms and equality cases, of Hamamoto's one-dimensional inequality and of his sharp uncertainty principle for solenoidal fields.

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arXiv2026

A Fourier approach to the sharp stability of Heisenberg Uncertainty Principles of higher and fractional orders: a new perspective

We show that, on the Fourier side, the sharp second order Heisenberg Uncertainty Principle (HUP) and its sharp stability for functions are nothing but the first order $L^{2}$-Caffarelli--Kohn--Nirenberg inequality and its sharp stability, applied to their Fourier transforms. This gives a proof in a few lines of the sharp stability estimate that we established recently by a much longer argument. The same observation then produces two one-parameter families of sharp fractional Heisenberg Uncertainty Principles, in which the gradient is replaced by a fractional power of the Laplacian: for each of them we compute the sharp constants, characterize all the optimizers, and establish the sharp stability estimates. The stability constant of the first family is equal to $1$ for every nonnegative order and every dimension. Negative orders, where the fractional Laplacian becomes a Riesz potential, are treated as well. Finally, we show that the deficit carries much more information than the distance to the optimizers alone: it controls an explicit chain of successive remainder terms whose constants are the successive spectral gaps of an explicit operator. For the classical Heisenberg Uncertainty Principle this gives three remainder terms with optimal explicit constants, and for the second order HUP a chain of four remainder terms measured against explicit confluent hypergeometric profiles and coupled through one single set of parameters. As applications, we also establish the chain of stability of the HUP for curl-free vector fields.

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arXiv2026

The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates

In this paper, we study the weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivatives. Firstly, we prove a family of the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities that complements the results in [{\it C. Cazacu, J. Flynn and N. Lam, Calc. Var. Partial Differential Equations 62 (2023), no. 4, Paper No. 118, 26 pp.}] and [{\it A. T. Duong and V. H. Nguyen, On the sharp second order Caffarelli-Kohn-Nirenberg inequality. Ann. Fenn. Math., 50(1):275--286, 2025}]. Secondly, we establish a stability version of the sharp weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields proved by Cazacu, Flynn and Lam. Finally, we prove a stability estimate for the sharp weighted second order $L^2$-Caffarelli-Kohn-Nirenberg inequalities established in this paper. Our approach is based on the spherical harmonic decomposition method, the one dimensional integral inequalities and their improvements.

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arXiv2026

Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincaré Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions

We introduce a new family of weighted Gaussian $L^2$-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli--Kohn--Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincaré inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.

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arXiv2024

A complete description of the asymptotic behavior at infinity of positive radial solutions to $Δ^2 u = u^α$ in $\mathbf R^n$

We consider the biharmonic equation $Δ^2 u = u^α$ in $\mathbf R^n$ with $n \geqslant 1$. It was proved that this equation has a positive classical solution if, and only if, either $α\leqslant 1$ with $n \geqslant 1$ or $α\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $α\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $α\leqslant 1$ with $n \geqslant 1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is higher order, we propose a new approach which relies on a representation formula and asymptotic analysis arguments. We believe that the approach introduced here can be conveniently applied to study other problems with higher order operators.

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arXiv2023

An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences

For $n > k \geq 0$, $λ>0$, and $p, r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{ |x-y|^λ|y"|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)} \] with $y = (y', y") \in \mathbf R^{n-k} \times \mathbf R^k$ under the two necessary conditions \[ β< \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 < λ\leq n-k,\\ & n - λ- k/r & & \text{if } \; n-k < λ, \end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { β+ λ} n = 2 -\frac kn. \] We call this the optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$. The existence of an optimal pair for this new inequality is also studied. The motivation of working on the above inequality is to provide a unification of many known Hardy-Littewood-Sobolev inequalities including the classical Hardy-Littewood-Sobolev inequality when $k=β=0$, the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ when $k=1$ and $β= 0$, and the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ with extended kernel when $k=1$ and $β\ne 0$. We show that the above condition for $β$ is sharp. In the unweighted case, namely $β=0$, our finding immediately leads to the sharp Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ with the optimal range $$0<λ<n-k/r,$$ which has not been observed before, even in the case $k=1$. As one of many consequences, we give a short proof of the Stein-Weiss inequality in the context of $\mathbf R^{n-k} \times \mathbf R^n$.

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arXiv2022

The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle

In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequalities which contains the sharp second order uncertainty principle recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case. We also show the sharpness of our inequalities for several classes of parameters. Finally, we prove two stability versions of the sharp second order uncertainty principle of Cazacu, Flynn and Lam by showing that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ norm of gradient of functions.

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arXiv2020

Extremals for the Singular Moser-Trudinger Inequality via n-Harmonic Transplantation

The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $Ω\subset\mathbb{R}^n$ is bounded, $ω_{n-1}$ is the $\mathcal{H}^{n-1}$ measure of the unit sphere, then for $α>0$ and $β\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_Ω\frac{e^{α|u|^{n/(n-1)}}}{|x|^β}\leq C \ \Leftrightarrow \ \fracα{α_n}+\fracβ{n}\leq1,\qquad $$ where $α_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(Ω) \ | \ \int_Ω |\nabla u |^n \leq1 \right\}$. We prove that the supremum is attained on any domain $Ω$. The paper also fills in the gaps in the proof of [Lin K.C., Extremal functions for Moser's inequality, \textit{Trans. of. Am. Math. Soc.}, 384 (1996), 2663--2671], which deals with the case $β=0.$

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arXiv2020

Liouville type theorems for fractional elliptic problems

In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-Δ)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a classification of stable solutions to the fractional Lane-Emden system $$\begin{cases} (-Δ)^s u = v^p \mbox{ in }\mathbb R^N (-Δ)^s v = u^q \mbox{ in }\mathbb R^N \end{cases}$$ with $p>1$ and $ q>1$. In our knowledge, this is the first classification result for stable solutions of the fractional Lane-Emden system in literature.

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arXiv2020

The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms

Let $2\leq m < n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ the Lorentz-Sobolev space of order $m$ in the hyperbolic space $\mathbb H^n$. In this paper, we establish the following Adams inequality in the Lorentz-Sobolev space $W^m L^{\frac nm,q}(\mathbb H^n)$ \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}\leq 1} \int_{\mathbb H^n} Φ_{\frac nm,q}\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] for $q \in (1,\infty)$ if $m$ is even, and $q \in (1,n/m)$ if $m$ is odd, where $β_{n,m}^{q/(q-1)}$ is the sharp exponent in the Adams inequality under Lorentz-Sobolev norm in the Euclidean space. To our knowledge, much less is known about the Adams inequality under the Lorentz-Sobolev norm in the hyperbolic spaces. We also prove an improved Adams inequality under the Lorentz-Sobolev norm provided that $q\geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq \frac nm$ if $m$ is odd, \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}^q -λ\|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb H^n} Φ_{\frac nm,q}\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dV_g < \infty \] for any $0< λ< C(n,m,n/m)^q$ where $C(n,m,n/m)^q$ is the sharp constant in the Lorentz-Poincaré inequality. Finally, we establish a Hardy-Adams inequality in the unit ball when $m\geq 3$, $n\geq 2m+1$ and $q \geq 2n/(n-1)$ if $m$ is even and $2n/(n-1) \leq q \leq n/m$ if $m$ is odd \[ \sup_{u\in W^mL^{\frac nm,q}(\mathbb H^n),\, \|\nabla_g^m u\|_{\frac nm,q}^q -C(n,m,\frac nm)^q \|u\|_{\frac nm,q}^q \leq 1} \int_{\mathbb B^n} \exp\big(β_{n,m}^{\frac q{q-1}} |u|^{\frac q{q-1}}\big) dx < \infty. \]

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arXiv2020

The sharp Sobolev type inequalities in the Lorentz--Sobolev spaces in the hyperbolic spaces

Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p$ which generalizes the result in \cite{NgoNguyenAMV} to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincaré-Sobolev type inequalities in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p < n$ which generalize the results in \cite{NguyenPS2018} to the setting of Lorentz-Sobolev spaces. Finally, we provide the improved Moser-Trudinger type inequalities in $W^1L^{n,q}(\mathbb{H}^n)$ in the critical case $p= n$ with $1\leq q \leq n$ which generalize the results in \cite{NguyenMT2018} and improve the results in \cite{YangLi2019}. In the proof of the main results, we shall prove a Pólya--Szegö type principle in $W^1 L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p$ which maybe is of independent interest.

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arXiv2019

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ Δ^m u = \pm u^α \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $α\in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $α$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $α$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.

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arXiv2019

The sharp Hardy--Moser--Trudinger inequality in dimension $n$

In this paper, we prove a Hardy--Moser--Trudinger inequality in the unit ball $\mathbb B^n$ in $\mathbb R^n$ which improves both the classical singular Moser--Trudinger inequality and the classical Hardy inequality at the same time. More precisely, we show that for any $β\in [0,n)$ there exists a constant $C>0$ depending only on $n$ and $β$ such that \[ \sup_{u\in W^{1,n}_0(\mathbb B^n), \mathcal H(u) \leq 1}\int_{\mathbb B^n} e^{(1-\fracβn)α_n |u|^{\frac n{n-1}}} |x|^{-β} dx \leq C \] where $α_n = n ω_{n-1}^{\frac1{n-1}}$ with $ω_{n-1}$ being the surface area of the unit sphere $S^{n-1} = \partial \mathbb B^n$, and \[ \mathcal H(u) = \int_{\mathbb B^n} |\nabla u|^n dx -\left(\frac{2(n-1)}n\right)^n \int_{\mathbb B^n} \frac{|u|^n}{(1-|x|^2)^n} dx. \] This extends an inequality of Wang and Ye in dimension two to higher dimensions and to the singular case as well. The proof is based on the method of transplantation of Green's functions and without using the blow-up analysis method. As a consequence, we obtain a singular Moser--Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev \cite[Conjecture $5.2$]{MST}. We also propose an inequality which extends the singular Hardy--Moser--Trudinger inequality to any bounded convex domain in $\mathbb R^n$ which is analogue of the conjecture of Wang and Ye in higher dimensions.

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arXiv2019

Supercritical Moser-Trudinger inequalities and related elliptic problems

Given $α>0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (α_n + |x|^α) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( α_n |u|^{\frac{n}{n-1} + |x|^α} \big) dx < +\infty, \] where $W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the class of functions $W^{1,n}_{0,{\rm rad}}(B)$, we should emphasize that the above inequalities fail in $W^{1,n}_{0,{\rm rad}}(B)$. Questions concerning the sharpness of the above inequalities as well as the existence of the optimal functions are also studied. To illustrate the finding, an application to a class of boundary value problems on balls is presented. This is the second part in a set of our works concerning functional inequalities in the supercritical regime.

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arXiv2019

A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space $L_{2^\star + |x|^α} (B)$, $2^\star = 2n/(n-2)$, $α>0$, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequality \[ \sup\Big\{\int_B |u(x)|^{2^\star+|x|^α} dx : u\in H^1_{0,{\rm rad}}(B), \|\nabla u\|_{L^2(B)} =1\Big\} < +\infty \] holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[ H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^α} (B) \] with $2\leq m < n/2$, $2_m^* = 2n/(n-2m)$, and $α>0$ holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.

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arXiv2019

The Leray--Adams inequality

In this paper, we establish the following Leray--Adams type inequality on a bounded domain $Ω$ in $\mathbb R^{4}$ containing the origin, \[ \sup_{u\in C_0^\infty(Ω), \tilde I_4[u,Ω,R] \leq 1} \int_Ω\exp\left(c\left( \frac{|u|}{E_2^β\left(\frac{|x|}R\right)}\right)^2\right) dx \leq C |Ω| \] for some constants $c >0$ and $C >0$, where $β\geq 1$, $R \geq \sup_{x\in Ω} |x|$, $ \tilde I_4[u,Ω,R]:= \int_Ω|Δu|^2 dx - \int_Ω\frac{|u|^2}{|x|^{4} E_1^2\left(\frac{|x|}R\right)} dx, $ and $E_1(t) = 1-\ln t$, $E_2(t) = \ln (eE_1(t))$ for $t \in (0,1]$. This extends the Leray--Trudinger inequality recently established by Psaradakis and Spector \cite{PS2015} and Mallick and Tintarev \cite{MT2018} to the case of Laplacian operator. In the higher dimensions or higher order derivatives, we prove the Leray--Adams type inequality for radial function on the ball $B_r$ (with center at origin and radius $r >0$) in $\mathbb R^n$.

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