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Thanh Viet Phan

Publications and source records attributed to Thanh Viet Phan.

8 recordsLinked to original sources
arXiv2022

Ground state solution of a Kirchhoff type equation with singular potentials

We study the existence and blow-up behavior of minimizers for $E(b)=\inf\Big\{\mathcal{E}_b(u) \,|\, u\in H^1(R^2), \|u\|_{L^2}=1\Big\},$ here $\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\mathcal{E}_b(u)= \int_{R^2} |\nabla u|^2 dx+ b(\int_{R^2} |\nabla u|^2d x)^2+\int_{R^2} V(x) |u(x)|^2 dx - \frac{a}{2} \int_{R^2} |u|^4 dx,$ where $a>0$ and $b>0$ are constants. When $V(x)= -|x|^{-p}$ with $0<p<2$, we prove that the problem has (at least) a minimizer that is non-negative and radially symmetric decreasing. For $a\ge a^*$ (where $a^*$ is the optimal constant in the Gagliardo-Nirenberg inequality), we get the behavior of $E(b)$ when $b\to 0^+$. Moreover, for the case $a=a^*$, we analyze the details of the behavior of the minimizers $u_b$ when $b\to 0^+$.

math.AP↗
arXiv2019

Ground state of the mass-critical inhomogeneous nonlinear Schrodinger functional

We study the ground state problem of the nonlinear Schrodinger functional with a mass-critical inhomogeneous nonlinear term. We provide the optimal condition for the existence of ground states and show that in the critical focusing regime there is a universal blow-up profile given by the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.

math-ph↗
arXiv2018

Blow-up profile of the focusing Gross-Pitaevskii minimizer under self-gravitating effect

We consider a Bose-Einstein condensate in a 2D dilute Bose gas, with an external potential and an interaction potential containing both of the short-range attractive self-interaction and the long-range self-gravitating effect. We prove the existence of minimizers and analyze their behavior when the strength of the attractive interaction converges to a critical value. The universal blow-up profile is the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.

math-ph↗
arXiv2018

Blow-up for biharmonic Schrodinger equation with critical nonlinearity

We consider the minimizers for the biharmonic nonlinear Schrödinger functional $$ \mathcal{E}_a(u)=\int_{\mathbb{R}^d} |Δu(x)|^2 d x + \int_{\mathbb{R}^d} V(x) |u(x)|^2 d x - a \int_{\mathbb{R}^d} |u(x)|^{q} d x $$ with the mass constraint $\int |u|^2=1$. We focus on the special power $q=2(1+4/d)$, which makes the nonlinear term $\int |u|^q$ scales similarly to the biharmonic term $\int |Δu|^2$. Our main results are the existence and blow-up behavior of the minimizers when $a$ tends to a critical value $a^*$, which is the optimal constant in a Gagliardo--Nirenberg interpolation inequality.

math-ph↗
arXiv2017

Blow-up profile of Bose-Einstein condensate with singular potentials

The paper is concerned with the Bose-Einstein condensate described by the attractive Gross-Pitaevskii equation in R 2 , where the external potential is unbounded from below. We show that when the interaction strength increases to a critical value, the Gross-Pitaevskii minimizer collapses to one singular point and we analyze the details of the collapse exactly up to the leading order.

math.AP↗
arXiv2015

Kirszbraun extension on connected finite graph

We prove that the tight function introduced Sheffield and Smart (2012) is a Kirszbraun extension. In the real-valued case we prove that Kirszbraun extension is unique. Moreover, we produce a simple algorithm which calculates efficiently the value of Kirszbraun extension in polynomial time.

math.FA↗
arXiv2014

Some results of the Lipschitz constant of 1-Field on $\mathbb{R}^n$

We study the relations between the Lipschitz constant of $1$-field and the Lipschitz constant of the gradient canonically associated with this $1$-field. Moreover, we produce two explicit formulas that make up Minimal Lipschitz extensions for $1$-field. As consequence of the previous results, for the problem of minimal extension by continuous functions from $\mathbb{R}^m$ to $\mathbb{R}^n$, we also produce analogous explicit formulas to those of Bauschke and Wang. Finally, we show that Wells's extensions of $1$-field are absolutely minimal Lipschitz extension when the domain of $1$-field to expand is finite. We provide a counter-example showing that this result is false in general.

math.FA↗