arXiv subjects
Publications and source records attributed to Juncheng Wei.
We prove the monotonicity property of the Robin torsion function in a smooth planar domain $Ω$ with a line of symmetry, provided that the Robin coefficient $β$ is greater than or equal to the negative of the boundary curvature $κ$, i.e., $β\geq - κ$ on $\partialΩ$. We also construct a counterexample showing that the curvature condition cannot, in general, be removed.
As is well-known, the solution of the Patlak--Keller--Segel system in 3D may blow up in finite time regardless of any initial cell mass. In this paper, we are interested in the suppression of blow-up and the critical mass threshold for the 3D Patlak--Keller--Segel--Navier--Stokes system via the Couette flow $(Ay, 0, 0)$. It is proved that if the Couette flow is sufficiently strong ($A$ is large enough), the initial cell mass is less than $16π^{2}$, and the zero modes of the initial velocity are sufficiently small, then the solutions for the system are global in time. The threshold $16π^2$ seems to be optimal: for every finite \(A>0\) and every \(M>16π^2\), we show that there exist smooth positive initial densities of mass \(M\), with zero initial velocity perturbation, whose corresponding solutions blow up in finite time. A key ingredient is a time-integrable decay estimate for the nonconstant part of the zero-mode velocity, which allows the logarithmic Hardy--Littlewood--Sobolev inequality to be used throughout the range $\frac{M}{2π}<8π$. Moreover, the global existence is obtained by combining quasi-linear space-time estimates proposed by Wei--Zhang (Comm. Pure Appl. Math., 2021) and a free-energy argument as in Bedrossian--He (SIAM J. Math. Anal., 2017) for the zero mode. The blow-up result follows from an exact invariant zero-mode reduction to the two-dimensional Patlak--Keller--Segel system and a localized virial argument.
We prove uniqueness, up to phase rotations and translations, of the ground state for the $L^2$-critical boson star equation \[ (\sqrt{-Δ}+1)Q =\bigl(|x|^{-1}* |Q|^2\bigr)Q, \qquad x\in\mathbb R^3. \] We also determine the kernels of the real and imaginary linearized operators. As a consequence, we show the Lieb-Yau uniqueness conjecture holds when the mass is sufficiently close to the Chandrasekhar mass.
We prove that every bounded stable entire solution $v$ of the Allen--Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.
Let $\Th(α,L)=\sum_{v\in L}e^{-πα|v|^2}$ for $α>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}|v|^{-2s}$ for $s>3/2$ be the theta and Epstein zeta functions associated to the lattice $L$, respectively. We are particularly interested in physically relevant dimension three. Fix the covolume of the lattice $L$ to $1$. Up to an orthogonal transformation, we prove that \begin{equation}\nonumber \argmin_{|L|=1}\Th(α,L)= \begin{cases} \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;&\text{if}\;\; α>1, \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{if}\;\; α<1, \boldsymbol{\mathrm{FCC}}\;\;\mathrm{or}\;\; \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice},\;\;&\text{if}\;\;α=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\text{if}\;\; s>3/2. \end{equation} Therefore, we prove the Sarnak--Strömbergsson conjecture \cite[Inequalities (43)--(44), Section 5]{SS}.
We study bounded entire solutions of the Allen-Cahn equation in dimension three with quadratic energy growth. We prove that such a solution of finite Morse index I has at most 2I ends. Furthermore, we show that Morse index one solutions have two ends and are axially symmetric, which generalizes an earlier result by Florit-Simon.
We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.
For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross--Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mariş (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(ψ)\le C_J\bigl(I_c(ψ)+\ind(ψ)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.
We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in $\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.
In this paper, we consider the anisotropic Allen-Cahn equation $-\operatorname{div} a(Du)+W'(u)=0$ in $\mathbb{R}^N$, where $a(p):=DH(p)$ with $H(p)=\frac{1}{2}F(p)^2$ and $F$ a uniformly elliptic integrand, and $W(u)=\frac{1}{4}(1-u^2)^2$. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for $N\geq4$ in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for $N\geq5$ that is monotone in one direction but is not one-dimensional.
We exhibit an explicit closed seven-dimensional Riemannian manifold \[ (M,g)=\mathbb S^2(1)\times \mathbb S^5\left(\frac1{100}\right), \] where the displayed parameters denote sectional curvatures, for which \(\Ric_g>0\), and hence \(Q_g^{(2)}>0\). Moreover, \[ Q^{(4)}_g>0,\qquad Q^{(6)}_g>0, \] and the sixth-order GJMS operator \(P_{6,g}\) is strictly positive as a self-adjoint operator, but nevertheless \(P_{6,g}\) fails the strong maximum principle. The failure is caused by a nonconstant positive eigenvalue of \(P_{6,g}\) lying strictly below the eigenvalue of the constant mode. The example also has \(Y_2(M,[g])>0\) and \(Y_4(M,[g])>0\), while \(P_{6,g}\) does not have a positive Green function. It disproves Conjecture~1 of Andrade, Piccione, and Wei and its general-order formulation by Case and Gover.
We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg--Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in the whole plane. Multiplying a disk competitor by $F/f$ enables us to use the known minimality of the whole-plane vortex without changing the boundary trace. The difference of the two energies can be decomposed into Fourier modes. Every nonzero mode is nonnegative, and the zero mode is then handled by a Picone type identity.
In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in $\mathbb{R}^2$. More precisely, assuming that a bounded uniformly convex domain $Ω\subset \mathbb{R}^2$ has a connected boundary of class $C^{2,ε}$ with $ε\in (0,1)$, we show that if, for some nonzero constant $c_D$, the overdetermined elliptic problem \begin{equation} -Δu = αu \ \text{ in } \ Ω, \qquad u = 0 \ \text{ on } \ \partialΩ, \qquad \frac{\partial u}{\partial ν} = c_{D} \ \text{ on } \ \partialΩ\nonumber \end{equation} admits a nontrivial solution corresponding to a large eigenvalue $α$, then the domain $Ω$ must be a disk. Similarly, we establish that if a domain $Ω\subset \mathbb{R}^2$ has a connected Lipschitz boundary and the problem \begin{equation} -Δu = αu \ \text{ in } \ Ω, \qquad \frac{\partial u}{\partial ν} = 0 \ \text{ on } \ \partialΩ, \qquad u = c_{N} \ \text{ on } \ \partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a large eigenvalue $α$, then $Ω$ is a disk as well.
We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, Beckner's inequality \[ \fracα{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] holds for every $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer $N\geq 3$.
We study positive stable solutions of the supercritical Lane--Emden equation in the first Joseph--Lundgren interval. For a family of dimensions, we construct nonradial stable entire solutions with exponent close to the upper endpoint of this interval. This disproves a radiality conjecture of Chan and Wei. The construction begins with a smooth positive nonconstant solution on the sphere, which is obtained by matching a polar cap to an inner neck. A sharp expansion of the lowest shifted eigenvalue proves that the resulting singular cone is strictly stable. Finally, a minimal-solution and rescaling argument replaces the cone by a smooth stable entire solution while preserving its sphere variation. To the best of our knowledge, this is the first nontrivial example of nonradial stable solutions for the Emden-Fowler equation.
We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \fracα{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\ln\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}(\mathbb{S}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted $\ell ^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\geq 3$.
We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ αP_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ α\ge\frac{1}{2}$ and $α\not =1$. The proof starts from a Gegenbauer expansion and its weighted $\ell ^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\lfloor G\rfloor^2$ using a specially designed integration-by-parts identity for general even integer $N\geq 4$. For $N\in\{3,5,7\}$, the seminorm estimates are established via a descent to a three--dimensional nonlocal difference quotient identity.
We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.