arXiv subjects

Changfeng Gui

Publications and source records attributed to Changfeng Gui.

At least 19 recordsLinked to original sources

arXiv2026

Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data

We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in $L^2_{loc}(\mathbb{R}^2\setminus\{0\})$. For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the $L^2$-norm of the initial datum on a fundamental annulus with the Dirichlet energy of its caloric lift over one period in logarithmic time. This, together with a structural cancellation between the linear profile and the nonlinear remainder over one period in logarithmic time, yields a uniform $\dot{H}^1$-bound for the remainder, where the natural $L^2_{loc}$-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no $L^2$-coercivity, we recover uniform energy control for the remainder by exploiting $L^p$-estimates ($1<p<2)$, together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to $C^2_{loc}(\mathbb{R}^2\setminus\{0\})$. These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to $L^2_{\rm loc}$ data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.

math.AP↗

arXiv2026

Rigidity of Stationary Navier-Stokes Solutions with Transverse bound in Dimensions Five and Higher

For $n\geq5$, let $(u,p)$ be a smooth solution of the stationary incompressible Navier-Stokes equations in $\R^n\setminus\{x'=0\}$, where $\{x'=0\}$ is the $x_n$-axis. We prove that the scale-invariant transverse bound $|u(x)|\leq C |x'|^{-1}$ forces $u\equiv0$. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of $u$ and $\nabla u$ in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-Růžička weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case $|u(x)|\leq C |x|^{-1}$ in [1], yields the same expansion in exterior domains under the condition $|u(x)|\leq C |x'|^{-1}$.

math.AP↗

arXiv2026

Global Strict Monotonicity and Asymptotics of the First Steklov Eigenvalue for Regular Polygons

For a bounded Lipschitz planar domain $Ω$, let $σ_1(Ω)$ denote its first nonzero Steklov eigenvalue. Let $Ω_N$ be the regular $N$-gon normalized to have perimeter $2π$. We prove that \[ σ_1(Ω_{N+1})>σ_1(Ω_N),\qquad N\ge3. \] Consequently, $σ_1(Ω_N)$ increases strictly to the disk value $1$. We also show that the real first eigenspace has dimension two and derive the asymptotic expansion \[ σ_1(Ω_N) =1-\frac{2ζ(3)}{N^3} -\frac{8ζ(4)}{N^4} -\frac{26ζ(5)}{N^5} +O(N^{-6}). \] The proof combines an equivariant Schwarz--Christoffel pullback, a nodal selection of the two critical Fourier residue classes, and a positivity-improving Perron comparison for the associated reciprocal operators. The asymptotic expansion follows independently from a Schur reduction and the evaluation of Euler-type sums.

math.AP↗

arXiv2026

The Pólya--Szegő conjecture for convex polygons with many sides

For every sufficiently large N, we prove that the regular N-gon uniquely minimizes the first Dirichlet eigenvalue among convex polygons of prescribed area with at most N sides, up to rigid motions. Quantitative Faber--Krahn stability localizes minimizers near the disk but leaves their discrete geometry undetermined. We encode that geometry by a centered measure of exterior angles and prove an inverse-cubic Fourier inequality whose equality case is the uniform root configuration. Its defect gives a lower bound for the quadratic spectral excess over the regular polygon. Cyclic cancellation and a common material-coordinate comparison make the nonlinear difference small relative to the same defect. Exact constraint coordinates extend this comparison to arbitrarily small positive defects. Minimality then forces zero defect, which characterizes the regular polygon.

math.AP↗

arXiv2026

Planar Gross--Pitaevskii traveling waves at every subsonic speed

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.

math.AP↗

arXiv2026

Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on $\mathbb{S}^2$

In this paper, we establish symmetry results for solutions of the mean field equation \[ \fracα{2} Δu + e^u - 1 = 0 \] on $ \mathbb{S}^2 $ for $\frac{1}{3}\leq α\leq 1$, under a geometric condition $(\mathcal{H})$ introduced below. The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on $ \mathbb{S}^2 $. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.

math.AP↗

arXiv2026

Symmetry and critical points of second Neumann eigenfunctions on isosceles trapezoids and kites

In this paper, we determine the symmetry properties and non-vertex critical points of the second Neumann eigenfunction $u$, as well as the multiplicity of the corresponding eigenvalue, on isosceles trapezoids and kites. By exploiting reflection symmetry, we reduce the problem to a comparison between the second Neumann eigenvalue and the first mixed Dirichlet--Neumann eigenvalue on the half-domain. More precisely, for isosceles trapezoids with base angle $α\leq \fracπ{3}$, the second eigenfunction is antisymmetric. If $\fracπ{3}<α<\fracπ{2}$, there exists a critical height $\hat{h}(α)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(α)$ and symmetric when $h>\hat{h}(α)$, while at $h=\hat{h}(α)$ the second Neumann eigenvalue has multiplicity two. For a convex kite $P_1P_2P_3P_4$, where $P_1=(0,0)$, $P_2=(a,-h)$, $P_3=(1,0)$, and $P_4=(a,h)$, an analogous result holds: there exists a critical height $\tilde{h}(a)$ such that $u$ is symmetric with respect to the $x$-axis when $h<\tilde{h}(a)$ and antisymmetric with respect to the $x$-axis when $h>\tilde{h}(a)$, while at $h=\tilde{h}(a)$ the second Neumann eigenvalue has multiplicity two. In all cases where the second Neumann eigenvalue is simple, we determine all non-vertex critical points of the corresponding eigenfunction, thereby verifying the hot spots conjecture.

math.AP↗

arXiv2026

A Quantitative Pólya--Szegő Theorem for Tangential Polygons

For a bounded Lipschitz domain $Ω\subset\mathbb R^2$, let $T(Ω)=\int_Ωu_Ω\, dx$ denote its torsional rigidity, where $-Δu_Ω=1$ in $Ω$ and $u_Ω=0$ on $\partialΩ$. We prove a quantitative Pólya--Szegő inequality for tangential polygons. Let $N\ge3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot)$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(π/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya--Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler--Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber--Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.

math.AP↗

arXiv2026

Optimal Rigidity Results for the $k$-Hessian Equation of Lane--Emden Type

In this paper, we establish optimal Liouville theorems and classification results for the $k$-Hessian Lane--Emden equation \[ σ_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{Γ_k},\qquad u\geq 0, \] where $2\leq k<\frac{n}{2}$ and $p>0$. Let $p_- = \frac{nk}{n-2k}$ and the critical Hessian--Sobolev exponent $p_* = \frac{(n+2)k}{n-2k}$. Phuc and Verbitsky proved nonexistence of positive solutions for $k 2k$, without any additional assumption. For the limiting case $n=2k$, we classify finite-mass solutions to the $\frac{n}{2}$-Hessian Liouville equation under a proper asymptotic condition $u(x)\rightarrow-\infty$ as $|x|\rightarrow\infty$. In particular, we provide the fully nonlinear counterparts of the classical Liouville and classification theorems of Gidas--Spruck, Gidas--Ni--Nirenberg, and Caffarelli--Gidas--Spruck.

math.AP↗

arXiv2026

Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water

In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite cylinder, where the vorticity may have jump discontinuities across internal streamlines. The analysis is carried out by applying a hodograph transformation, which reformulates the original free boundary problem into an abstract elliptic boundary value problem. Compared to previously studied steady water waves, the present setting introduces several novel features: the presence of an internal interface, an unbounded spatial domain, and a non-Fredholm linearized operator. To address these difficulties, we introduce a height function formulation, casting the problem as a transmission problem with suitable transmission conditions. A singular bifurcation approach is then employed, combining global bifurcation theory with Whyburn's topological lemma. The singular limit is taken in a natural deep-water topology based on the surface trace and the derivatives of the height function. An exact mean-flux identity controls the closing zero Fourier mode uniformly as the regularization is removed. This yields a connected global continuum of exact waves together with explicit norm, ellipticity, surface-obliqueness, and parameter-boundary alternatives.

math.AP↗

arXiv2026

A positive answer to the generalized Chang-Yang conjecture on $\mathbb{S}^N$

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 $N\geq 3$.

math.AP↗

arXiv2026

Sharp Beckner's Inequalities for Axially Symmetric Functions on $\mathbb{S}^N$

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 $N\geq 3$.

math.AP↗

arXiv2026

Axially Symmetric Rigidity for a $Q$-Curvature-Type Equation on $ \mathbb{S}^N $

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.

math.AP↗

arXiv2026

Non-radial solutions for the quasi-linear Hénon type $N$-Laplacian Liouville equation

In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \begin{equation*}\label{0} -Δ_N u=|x|^{Nα}e^{u}, \qquad x\in \R^N, \end{equation*} where $N \geq 2$. For $\al>0$, by carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $α$ equals to the critical values $α(k):=\frac{\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{α(k)}$) to the above quasi-linear Hénon type Liouville equation such that $u\sim \ln|x|$, $|\nabla u|= O(|x|^{-1})$ at $\infty$ and $\int_{\R^N}|x|^{Nα}e^{u}\md x=N\left(\frac{N^2}{N-1}\right)^{N-1}(α+1)^{N-1}ω_N$. One should note that, $α(k)=k-1$ for $k\geq2$ when $N=2$. Our results successfully extend the existence result of J. Prajapat and G. Tarantello in \cite{PT} concerning the $2$-dimension and Laplacian case (i.e., $N=2$) to the more general $N$-dimension and $N$-Laplacian cases ($N\geq 2$), and extend the results of F. Gladiali, M. Grossi, and S. L. N. Neves in \cite{GGN} and the authors in \cite{DDGL} from $1<p<N$ to the much more complicated limiting case $p=N$. We introduced some new ideas and overcame a series of crucial difficulties, including the nonlinearity nature of the $N$-Laplacian $Δ_N$, the lack of Green integral representation formula and critical weighted Sobolev embedding inequality, the absence of Kelvin type transforms for linearized/difference equations, the invariance of the total mass under scalings of $u$, and the signs-changing and divergence (to $-\infty$) at $\infty$ of the solutions, which makes the suitable choices of the approximate problems, the (normalized) approximate function sequences and the working space to be quite difficult.

math.AP↗

arXiv2026

Sharp weighted Carleman and Huber isoperimetric inequalities on the unit ball in higher dimensions

In this paper, using a limiting approach, we establish a new type of weighted Carleman inequality in all dimensions $n\geq 2$ and classify all extremal functions. In particular, when $n=2$, we prove that our inequality is equivalent to a sharp norm inequality in the Bergman space. In even dimensions, we further establish a sharp weighted Huber isoperimetric inequality on the unit ball, which generalizes Huber's original result \cite[Ann. Math., 1954]{Huber} and may be regarded as a sharp counterpart of Y. Wang's isoperimetric inequality in the unit ball \cite[Adv. Math., 2015]{Wang}.

math.DG↗

arXiv2026

On two-dimensional steady compactly supported Euler flows with constant vorticity

In this paper, we study the two-dimensional steady compactly supported incompressible Euler equations with free boundaries. We consider flows with constant vorticity that are perturbations of annular equilibria, in contrast to the laminar flows that predominate in the existing literature on steady water waves. More precisely, we analyze three distinct classes of steady Euler flows with compact support, which correspond, respectively, to partially overdetermined, two-phase overdetermined, and (fully) overdetermined elliptic free-boundary problems. Our main contributions are threefold. For each class, we first prove a flexibility result-the existence of nontrivial admissible domains-by combining shape derivatives with local bifurcation theory. Second, we establish the corresponding rigidity results. Third, we apply the implicit function theorem to show that the standard annular flows are stable under small perturbations of the Neumann boundary condition. These results provide new perspectives on the theory of overdetermined elliptic problems.

math.AP↗

arXiv2026

Symmetry breaking for the complex sine-Gordon equation

We consider the existence of vortex solutions to the complex sine-Gordon II (CSG2) equation, which can be viewed as an analogy of the Ginzburg-Landau (GL) equation. Using the nontrivial kernels $η_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $Ψ_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $Ψ_{2,α}$. Explicit formulas of these solutions are also available, from which we propose a new bilinear system for this equation. Our method can be generalized to higher degree case. Nondegenracy and stability of degree-1 solution are also proved. Finally, we formally discuss the Lyapunov-Schmidt reduction procedure for the multivortex solutions of the CSG2 and GL equation.

math.AP↗

arXiv2026

Mixed Torsion on Right Triangles and the Pólya--Szegő Monotonicity Problem for Regular Polygons

Motivated by the polygonal Pólya--Szegő conjecture for torsional rigidity, we study two monotonicity problems for torsional rigidity. The first concerns a mixed torsion problem on fixed-area right triangles, with a Dirichlet condition on one leg and Neumann conditions on the other leg and on the hypotenuse. We prove that the mixed torsional rigidity strictly increases as the ratio of the Neumann leg to the Dirichlet leg increases. The proof uses a Hadamard shape derivative, a Pohozaev-type identity, and a monotonicity result for the mixed torsion function. We also prove a similar result for the mixed ground state of Laplacian. The second concerns regular polygons. If $P_N$ denotes the regular $N$-gon of area $π$, we prove, by a purely analytic Schwarz--Christoffel/Bergman analytic-content argument, that \[ T^D(P_{N+1})>T^D(P_N),\qquad N\ge3, \] where $T^D$ is the Dirichlet torsional rigidity. We also obtain the asymptotic expansion \[ T^D(P_N)=\fracπ{8}-\frac{πζ(3)}{N^3} +\frac{π^5}{45N^4}+O(N^{-5}). \]

math.AP↗