Searcharxiv⌕ Search

arXiv subjects

Juncheng Wei

Publications and source records attributed to Juncheng Wei.

At least 127 records · Page 7Linked to original sources
arXiv2023

Complete metrics with constant fractional higher order $Q$-curvature on the punctured sphere

This manuscript is devoted to constructing complete metrics with constant higher fractional curvature on punctured spheres with finitely many isolated singularities. Analytically, this problem is reduced to constructing singular solutions for a conformally invariant integro-differential equation that generalizes the critical GJMS problem. Our proof follows the earlier construction in Ao {\it et al.} \cite{MR3694645}, based on a gluing method, which we briefly describe. Our main contribution is to provide a unified approach for fractional and higher order cases. This method relies on proving Fredholm properties for the linearized operator around a suitably chosen approximate solution. The main challenge in our approach is that the solutions to the related blow-up limit problem near isolated singularities need to be fully classified; hence we are not allowed to use a simplified ODE method. To overcome this issue, we approximate solutions near each isolated singularity by a family of half-bubble tower solutions. Then, we reduce our problem to solving an (infinite-dimensional) Toda-type system arising from the interaction between the bubble towers at each isolated singularity. Finally, we prove that this system's solvability is equivalent to the existence of a balanced configuration.

math.AP↗
arXiv2023

Nonuniqueness results for constant sixth order $Q$-curvature metrics on spheres with higher dimensional singularities

We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order GJMS operator, our objective is two-fold. First, we use a classical bifurcation technique to prove that there exists infinitely many constant $Q$-curvature metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$. As a by-product, we find the sixth order Yamabe invariant on this product manifold can be arbitrarily close to that of the round dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant. Second, when the underlying manifold is noncompact, we apply a bifurcation technique on Riemannian covering to construct infinitely many complete metrics with constant sixth order $Q$-curvature conformal to $\mathbb{S}^{n_1} \times \mathbb{R}^{n_2}$ or $\mathbb{S}^{n_1} \times \mathbb{H}^{n_2}$, where $n_1+n_2\geqslant 7$. Consequently, we obtain infinitely many solutions to the singular constant GJMS equation on round spheres $\mathbb{S}^n\setminus \mathbb{S}^k$ blowing up along a minimal equatorial subsphere with $0 \leqslant k<\frac{n-6}{2}$; this dimension restriction is sharp in the topological sense.

math.DG↗
arXiv2023

On Beckner's Inequality for Axially Symmetric Functions on $\mathbb{S}^6$

We prove that axially symmetric solutions to the $Q$-curvature type problem $$ αP_6 u + 120(1-\frac{e^{6u}}{\int_{\mathbb{S}^6} e^{6u}})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^6 $$ must be constants, provided that $ \frac{1}{2}\leq α<1$. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<α<\frac{1}{2}$, this result is sharp. This result closes the gap of the related results in \cite{GHW2022}, which proved a similar uniqueness result for $α\geq 0.6168$. The improvement is based on two types of new estimates: one is a better estimate of the semi-norm $\lfloor G\rfloor^2$, the other one is a family of refined estimates on Gegenbauer coefficients, such as pointwise decaying and cancellations properties.

math.AP↗
arXiv2023

Existence and stability of infinite time blow-up in the Keller-Segel system

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; Δu - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v =&\; (-Δ_{\R^2})^{-1} u := \frac 1{2π} \int_{R^2} \, \log \frac 1{|x-z|}\,u(z,t)\, dz, \\ & \qquad\ u(\cdot ,0) = u_0 \geq 0\quad\hbox{in } R^2. \end{aligned} \right. \end{equation} We consider the {\em critical mass case} $\int_{R^2} u_0(x)\, dx = 8π$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8π$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of \equ{ks0} is globally defined and blows-up in infinite time. As $t\to+\infty $ it has the approximate profile $$ u(x,t) \approx \frac 1{λ^2} \ch{U}\left (\frac {x-ξ(t)}{λ(t)} \right ), \quad \ch{U}(y)= \frac{8}{(1+|y|^2)^2}, $$ where $λ(t) \approx \frac c{\sqrt{\log t}}, \ ξ(t)\to q $ for some $c>0$ and $q\in \R^2$. This result answers affirmatively the nonradial stability conjecture raised in \cite{g}.

math.AP↗
arXiv2023

Compactness of singular solutions to the sixth order GJMS equation

We study compactness properties of the set of conformally flat singular metrics with constant, positive sixth order Q-curvature on a finitely punctured sphere. Based on a recent classification of the local asymptotic behavior near isolated singularities, we introduce a notion of necksize for these metrics in our moduli space, which we use to characterize compactness. More precisely, we prove that if the punctures remain separated and the necksize at each puncture is bounded away from zero along a sequence of metrics, then a subsequence converges with respect to the Gromov--Hausdorff metric. Our proof relies on an upper bound estimate which is proved using moving planes and a blow-up argument. This is combined with a lower bound estimate which is a consequence of a removable singularity theorem. We also introduce a homological invariant which may be of independent interest for upcoming research.

math.DG↗
arXiv2023

On lattice hexagonal crystallization for non-monotone potentials

Let $L =\sqrt{\frac{1}{\Im(z)}}\Big({\mathbb Z}\oplus z{\mathbb Z}\Big)$ where $z \in \mathbb{H}=\{z= x+ i y\;\hbox{or}\;(x,y)\in\mathbb{C}: y>0\}$ be the two dimensional lattices with unit density. Assuming that $α\geq1$, we prove that \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}|\mathbb{P}|^2 e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice. More generally we prove that for $α\geq 1$ \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}(|\mathbb{P}|^2-\frac{b}α) e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice for $b\leq\frac{1}{2π}$ and does not exist for $b>\frac{1}{2π}$. As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in \cite{Oreport, Bet2016, Bet2018, Bet2019AMP} and extend the main results in \cite{LW2022} on minima of difference of two theta functions.

math.AP↗
arXiv2023

Trichotomy dynamics of the 1-equivariant harmonic map flow

For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=π~\mbox{ and }~ v_0(r)\sim r^{1-γ} ~\mbox{ as }~ r\to+\infty, \quad γ>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\infty ([0,\infty))} \sim \begin{cases} t^{\frac{γ-2}{2}}\ln t ~&\mbox{ if }~ 1<γ<2,\\ 1 ~&\mbox{ if }~ γ=2,\\ \ln t ~&\mbox{ if }~ γ>2,\\ \end{cases} ~\mbox{ as }~ t\to +\infty. \end{equation*}

math.AP↗
arXiv2022

On Minima of Difference of Epstein Zeta Functions and Exact Solutions to Lennard-Jones Lattice Energy

Let $ζ(s,z)=\sum_{(m,n)\in\mathbb{Z}^2\backslash\{0\}}\frac{(\Im(z))^s}{|mz+n|^{2s}}$ be the Eisenstein series/Epstein Zeta function. Motivated by widely used Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{12}-(\fracσ{|\cdot|})^{6} \Big), \endaligned\end{equation} in physics, in this paper, we consider the following lattice minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(ζ(6,z)-bζ(3,z)\Big), \;\;b=\frac{1}{σ^6} \endaligned\end{equation} and completely classify the minimizers for all $b\in \R$. Our results resolve an open problem in Blanc-Lewin \cite{Bla2015}, and a conjecture by Bétermin \cite{Bet2018}. Furthermore, our method of proofs works for general minimization problem \begin{equation}\aligned\nonumber \min_{z\in\mathbb{H}}\Big(ζ(s_1,z)-bζ(s_2,z)\Big), \;\;s_1>s_2>1 \endaligned\end{equation} which corresponds to general Lennard-Jones potential \begin{equation}\aligned\nonumber \mathcal{V}(|\cdot|^2):=4\varepsilon\Big( (\fracσ{|\cdot|})^{2s_1}-(\fracσ{|\cdot|})^{2s_2} \Big),\;\;s_1>s_2>1. \endaligned\end{equation}

math.AP↗
arXiv2022

Classification for positive singular solutions to critical sixth order equations

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation of our equation to run a moving spheres technique, which proves that solutions are radially symmetric with respect to the origin. Thus, in Emden--Fowler coordinates, we can reduce our problem to the study of an sixth order autonomous ODE with constant coefficients. The main heuristics behind our arguments is that since all the indicial roots of the ODE operator are positive, it can be decomposed into the composition of three second order operators satisfying a comparison principle. This allows us to define a Hamiltonian energy which is conserved along solutions, from which we extract their qualitative properties, such as uniqueness, boundedness, asymptotic behavior, and classification.

math.AP↗
arXiv2022

Asymptotics for positive singular solutions to subcritical sixth order equations

We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. Initially, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden--Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables. The difficulty is that the cylindrical PDE in this coordinate system is nonautonomous. Nonetheless, we define an associated nonautonomous Pohozaev functional, which can be proved to be asymptotically monotone. In addition, we show a priori estimates for these two functionals, from which we extract compactness properties. With this ingredients, we can perform an asymptotic analysis technique to prove our main result.

math.AP↗
arXiv2022

On Fila-King Conjecture in Dimension Four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}.

math.AP↗
arXiv2022

Infinitely many nonradial positive solutions for multi-species nonlinear Schrödinger systems in ${\mathbb R}^N$

In this paper, we consider the multi-species nonlinear Schrödinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-Δu_j+V_j(x)u_j=μ_ju_j^3+\sum_{i=1;i\not=j}^dβ_{i,j} u_i^2u_j\quad\text{in }\bbr^N, &u_j(x)>0\quad\text{in } {\mathbb R}^N, &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quad j=1,2,\cdots,d,\endaligned\right. \end{equation*} where $N=2,3$, $μ_j>0$ are constants, $β_{i,j}=β_{j,i}\not=0$ are coupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials $V_j(x)$ and coupling parameters $\{β_{i,j}\}$, {\it without any symmetric assumptions on the limit case of the above system}. Our result, giving a positive answer to the conjecture in Pistoia and Vaira \cite{PV22} and extending the results in \cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$ and is {\it almost optimal} for the coupling parameters $\{β_{i,j}\}$.

math.AP↗
arXiv2022

Sharp stability of the logarithmic Sobolev inequality in the critical point setting

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{πd e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in W^{1,2}(\mathbb{R}^d)$ with $d\geq2$ and $\|u\|_{L^2(\mathbb{R}^d)}=1$. It is well known that extremal functions of this inequality are precisely the Gaussians \begin{eqnarray*} \mathfrak{g}_{σ,z}(x)=(πσ)^{-\frac{d}{2}}\mathfrak{g}_{*}\bigg(\sqrt{\fracσ{2}}(x-z)\bigg)\quad\text{with}\quad \mathfrak{g}_{*}(x)=e^{-\frac{|x|^2}{2}}. \end{eqnarray*} We prove that if $u\geq0$ satisfying $(ν-\frac12)c_0<\|u\|_{H^1(\mathbb{R}^d)}^2<(ν+\frac12)c_0$ and $\|-Δu+u-2u\log |u|\|_{H^{-1}}\leqδ$, where $c_0=\|\mathfrak{g}_{1,0}\|_{H^1(\mathbb{R}^d)}^2$, $ν\in \mathbb{N}$ and $δ>0$ sufficiently small, then \begin{eqnarray*} \text{dist}_{H^1}(u, \mathcal{M}^ν)\lesssim\|-Δu+u-2u\log |u|\|_{H^{-1}} \end{eqnarray*} which is optimal in the sense that the order of the right hand side is sharp, where \begin{eqnarray*} \mathcal{M}^ν=\{(\mathfrak{g}_{1,0}(\cdot-z_1), \mathfrak{g}_{1,0}(\cdot-z_2), \cdots, \mathfrak{g}_{1,0}(\cdot-z_ν))\mid z_i\in\bbr^d\}. \end{eqnarray*} Our result provides an optimal stability of the Euclidean logarithmic Sobolev inequality in the critical point setting.

math.AP↗
arXiv2022

The BCC lattice in a long range interaction system

While the hexagonal lattice is ubiquitous in two dimensions, the body centered cubic lattice and the face centered lattice are both commonly observed in three dimensions. A geometric variational problem motivated by the diblock copolymer theory consists of a short range interaction energy and a long range interaction energy. In three dimensions, and when the long range interaction is given by the nonlocal operator $(-Δ)^{-3/2}$, it is proved that the body centered cubic lattice is the preferred structure.

math.AP↗
arXiv2022

Nematic liquid crystal flow with partially free boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

math.AP↗
arXiv2022

Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth

We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}Δu + u = f(u){\text{ in }}Ω,{\text{ }}u > 0{\text{ in }}Ω,{\text{ }}u = 0{\text{ on }}\partial Ω, \] where $Ω$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial Ω\in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial Ω$.

math.AP↗