arXiv subjects

Changfeng Gui

Publications and source records attributed to Changfeng Gui.

At least 19 recordsLinked to original sources

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 Mari\c{s} (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\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

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

We prove that for every integer $N\geq 3$ and $\alpha\geq \frac{1}{2}$, Beckner's inequality \[ \frac{\alpha}{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] 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 $\alpha\geq \frac{1}{2}$, Beckner's inequality \begin{equation*} \frac{\alpha}{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

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 \[ \sigma_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{\Gamma_k},\qquad u\geq 0, \] 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

A Quantitative P\'olya--Szeg\H{o} Theorem for Tangential Polygons

For a bounded Lipschitz domain $\Omega\subset\mathbb R^2$, let $T(\Omega)=\int_\Omega u_\Omega\, dx$ denote its torsional rigidity, where $-\Delta u_\Omega=1$ in $\Omega$ and $u_\Omega=0$ on $\partial\Omega$. We prove a quantitative P\'olya--Szeg\H{o} 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 \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(\pi/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]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 P\'olya--Szeg\H{o} 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

Non-radial solutions for the quasi-linear H\'enon type $N$-Laplacian Liouville equation

In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \begin{equation*}\label{0} -\Delta_N u=|x|^{N\alpha}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 $\alpha$ equals to the critical values $\alpha(k):=\frac{\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{\alpha(k)}$) to the above quasi-linear H\'enon type Liouville equation such that $u\sim \ln|x|$, $|\nabla u|= O(|x|^{-1})$ at $\infty$ and $\int_{\R^N}|x|^{N\alpha}e^{u}\md x=N\left(\frac{N^2}{N-1}\right)^{N-1}(\alpha+1)^{N-1}\omega_N$. One should note that, $\alpha(k)=k-1$ 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 $\Delta_N$, 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

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 $\eta_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $\Psi_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $\Psi_{2,\alpha}$. 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 P\'olya--Szeg\H{o} Monotonicity Problem for Regular Polygons

Motivated by the polygonal P\'olya--Szeg\H{o} 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 $\pi$, 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 \[ T^D(P_N)=\frac{\pi}{8}-\frac{\pi\zeta(3)}{N^3} +\frac{\pi^5}{45N^4}+O(N^{-5}). \]

math.AP↗

arXiv2026

Liouville theorems for $p$-Laplacian equations in convex cones without finite-energy condition

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-\Delta ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot \nu =0 \quad\,\,\, &{\rm{on}} \,\, \partial\mathcal{C}, \end{aligned} \right. \end{equation*} where $N\geq3$, $1<p<N$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone and $\Delta ^{H}_{p}$ is the anisotropic Finsler $p$-Laplacian operator. If $f(u)$ is nonnegative and subcritical, we prove that every bounded nonnegative solution in $\mathcal{C}$ is identically zero. In particular, for $f(u)=u^{q}$ with $0<q<p^*-1$, we establish a pointwise decay estimate in $\mathcal{C}$ via the doubling argument and blowing-up method and prove that all nonnegative solutions must be zero without the boundedness assumption. Our results are the subcritical counterpart of the classification result for the critical case in \cite{CFR}, and extend the Liouville type theorems in $\mathbb{R}^{N}$ for the standard $p$-Laplacian in \cite{SZ} and for the anisotropic $p$-Laplacian in \cite{CFV, CHN} to general convex cones $\mathcal{C}$. In the critical case $f(u)=u^{p^*-1} $ and typical case $H(\xi)=|\xi|$, for $\frac{N+1}{3}<p<N$, we classified the positive solutions of the critical $p$-Laplacian equation in convex cones $\mathcal{C}$ without finite-energy assumption. This extends the classification result of \cite{Ou} in $\mathbb{R}^{N}$ to general convex cones $\mathcal{C}$, and removes the finite-energy assumption in \cite{CFR} in the typical case $H(\xi)=|\xi|$.

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 \[ \frac{\alpha}{2} \Delta u + e^u - 1 = 0 \] on $ \mathbb{S}^2 $ for $\frac{1}{3}\leq \alpha \leq 1$, 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 $\alpha\leq \frac{\pi}{3}$, the second eigenfunction is antisymmetric. If $\frac{\pi}{3}<\alpha<\frac{\pi}{2}$, there exists a critical height $\hat{h}(\alpha)$ at which the two symmetry branches cross: $u$ is antisymmetric when height $h<\hat{h}(\alpha)$ and symmetric when $h>\hat{h}(\alpha)$, while at $h=\hat{h}(\alpha)$ 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

Monotonicity of the first nonzero Steklov eigenvalue of regular $N$-gon with fixed perimeter

We study the first nontrivial Steklov eigenvalue of perimeter-normalized regular $N$-gons and show that it is strictly increasing in $N$. The proof mainly relies on an analytic framework that establishes a refined asymptotic expansion in three steps: first, identifying the Steklov eigenvalue as the maximal eigenvalue of a Toeplitz-type operator; second, deriving the eigenvalue and its associated eigenfunctions simultaneously via Schur reduction; and finally, obtaining the exact coefficients in the Schur moment expansion by evaluating Euler-type sums. The monotonicity is proved to be eventual, holding for $N\ge 20$. For the remaining cases $3\le N\le 20$, we provide complementary computer-assisted verification, confirming monotonicity across the full range of $N$.

math.AP↗

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

Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

In this paper, for general $n\geq2$, we classify solutions to $n$-Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space $\mathbb{R}^{n}_{+}$. Under the positive nonlinear Neumann boundary condition, our result extend the classification result for the second order Liouville equation in \cite{Li} from $n=2$ to general $n\geq2$, and also extend the classification result for critical $p$-Laplacian equation in \cite{Zhou} from $p<n$ to $p=n$.

math.AP↗

arXiv2026

A selection principle for 2D steady Euler flows via the vanishing viscosity limit

The 2D Euler system, which governs inviscid incompressible fluid flow, can admit infinitely many steady solutions in a given domain with slip boundary conditions. To select physical classical solutions, we investigate the vanishing viscosity limits of the steady Navier-Stokes system. The vanishing viscosity limits in periodic strips or bounded connected domains are completely characterized, even when strong boundary layers may appear. More precisely, we show that the only vanishing viscosity limits in a bounded connected domain are flows with constant vorticity. The significance of this result is that the approximating Navier-Stokes solutions are not required to have nested closed streamlines, an essential assumption in the century-old Prandtl-Batchelor theorem. For flows in an infinitely long strip, if the viscous velocity (but not the pressure) is periodic in the strip direction, we show that the only vanishing viscosity limits are constant flows, Couette flows, and Poiseuille flows. The proof relies on a delicate analysis of the streamlines for both viscous and inviscid flows, in which a key observation is that the set of chaotic streamlines for the Euler flow is null with respect to two-dimensional Lebesgue measure. The second result depends not only on the first but also on a powerful rigidity theorem that any non-shear steady classical Euler flow in a periodic strip must have closed streamlines, established via an analysis of streamlines and a novel total curvature estimate.

math.AP↗

arXiv2026

On the forward self-similar solutions to the two-dimensional Navier-Stokes equations

We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the major difficulty for the 2D problem is the criticality in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable, so that the classical local energy estimates are not available. Our key ideas are to decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part, and to exploit a kind of inherent cancellation relation between the linear part and the perturbation part. These, together with suitable choices of multipliers, enable us to control the interaction terms and to establish the $H^1$-estimates for the perturbation part. Furthermore, we can get an optimal pointwise estimate via investigating the corresponding Leray equations in weighted Sobolev spaces.This gives the faster decay of the perturbation part at infinity and compactness, which play important roles in proving the existence of global-in-time self-similar solutions.

math.AP↗

arXiv2025

Stability of Solitary Capillary-Gravity Water Waves in Three Dimensions

This paper establishes the conditional orbital stability of fully localized solitary waves for the three-dimensional capillary-gravity water wave problem in finite depth under strong surface tension. The waves, constructed via a non-variational Lyapunov-Schmidt reduction in [26], are not energy minimizers and thus require a direct stability analysis. We adapt the Grillakis-Shatah-Strauss framework within Mielke's approach to handle the mismatch between well-posedness and energy spaces. The proof relies on spectral analysis of the linearized dynamics and careful treatment of the Hamiltonian structure defined by the energy and momentum functionals.

math.AP↗