arXiv subjects

Yujin Guo

Publications and source records attributed to Yujin Guo.

43 records · Page 3Linked to original sources

arXiv2015

Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials

We are interested in the attractive Gross-Pitaevskii (GP) equation in $\R^2$, where the external potential $V(x)$ vanishes on $m$ disjoint bounded domains $Ω_i\subset \R^2\ (i=1,2,\cdots,m)$ and $V(x)\to\infty$ as $|x|\to\infty$, that is, the union of these $Ω_i$ is the bottom of the potential well. By making some delicate estimates on the energy functional of the GP equation, we prove that when the interaction strength $a$ approaches some critical value $a^*$ the ground states concentrate and blow up at the center of the incircle of some $Ω_j$ which has the largest inradius. Moreover, under some further conditions on $V(x)$ we show that the ground states of GP equations are unique and radially symmetric at leat for almost every $a \in (0, a^*)$.

math.AP↗

arXiv2014

Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

This paper is concerned with the properties of $L^2$-normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a>0$ approaches a critical value $a^*$, each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a\nearrow a^*$. However, when $a>0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

math.FA↗

arXiv2013

Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations

We study the following nonlinear Schrödinger equation $$ iu_t=-Δu+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:= \|Q\|_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\nearrow 2$.

math.AP↗

arXiv2013

On the Mass Concentration for Bose-Einstein Condensates with Attractive Interactions

We consider two-dimensional Bose-Einstein condensates with attractive interaction, described by the Gross-Pitaevskii functional. Minimizers of this functional exist only if the interaction strength $a$ satisfies $a < a^*= \|Q\|_2^2$, where $Q$ is the unique positive radial solution of $Δu-u+u^3=0$ in $\R^2$. We present a detailed analysis of the behavior of minimizers as $a$ approaches $a^*$, where all the mass concentrates at a global minimum of the trapping potential.

math-ph↗

arXiv2007

Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS

The singular parabolic problem $u_t=Δu -\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $λ$. As a byproduct, we prove that for sufficiently large $λ$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.

math.AP↗

arXiv2005

Compactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-Δu=\frac{λf(x)}{(1-u)^2}$ on a bounded domain $Ω\subset \R^N$, which models --among other things-- a simple electrostatic Micro-Electromechanical System (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for $1\leq N \leq 7$ on any domain $Ω$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^α$ can push back the critical dimension N=7 of this problem, by establishing compactness for the semi-stable branch on the unit ball, also for $N\geq 8$ and as long as $α>α_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for $λ$ in a natural range. In all these results, the conditions on the space-dimension and on the power of the profile are essentially sharp.

math.AP↗

arXiv2005

On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case

We analyze the nonlinear elliptic problem $Δu=\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage --represented here by $λ$-- is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value $λ^*$ (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers \cite{GG2} and \cite{GG3}. For now, we focus on the stationary equation where the challenge is to estimate $λ^*$ in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile $f$. Applying analytical and numerical techniques, the existence of $λ^*$ is established together with rigorous bounds. We show the existence of at least one steady-state when $λ< λ^*$ (and when $λ=λ^*$ in dimension $N< 8$) while none is possible for $λ>λ^*$. More refined properties of steady states --such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results-- are shown to depend on the dimension of the ambient space and on the permittivity profile.

math.AP↗