arXiv subjects

Hong-Ge Chen

Publications and source records attributed to Hong-Ge Chen.

9 recordsLinked to original sources

arXiv2026

On Brezis' open problem 2.2

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.

math.AP↗

arXiv2026

On Sirakov's equal-frequency uniqueness conjecture

Let $N\in\{2,3\}$, $0<\mu _1\leq\mu _2$, and $0<\beta<\mu _1$. We prove that the equal-frequency two-component cubic Schr\"odinger system \[ -\Delta u+u=\mu _1u^3+\beta uv^2, \qquad -\Delta v+v=\mu _2v^3+\beta u^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-\Delta w+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range. The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.

math.AP↗

arXiv2026

A Pfaffian Proof and Generalization of a Conjecture of Sun Zhiwei

Let $p$ be an odd prime, let $n=(p-1)/2$, and let $\chi=(\frac{\cdot}{p})$, with $\chi(0)=0$. For $a\in\mathbb F_p^\times$ define \[ D_a(x)=\det_{1\le i,j\le n}(x+\chi(i^2-aj)), \qquad D_a^{(0)}(x)=\det_{0\le i,j\le n}(x+\chi(i^2-aj)). \] We prove \[ D_a(0)=0 \quad\Longleftrightarrow\quad p\equiv 3 \pmod 4 \quad\text{and}\quad \chi(a n!)=1. \] For $p\equiv3\pmod4$ we also give explicit Pfaffian-square factorizations of $D_a(x)$ and $D_a^{(0)}(x)$. Let $s_p=(-1)^{\lfloor(p+1)/8\rfloor}$. If $\chi(a n!)=1$, then $s_pD_a(x)/x=s_pD_a^{(0)}(x)$ is a positive integer square. If $\chi(a n!)=-1$, then there is a positive integer $\sigma$ such that \[ s_pD_a(x)=\sigma^2(nx-1),\qquad s_pD_a^{(0)}(x)=-\sigma^2\bigl(n+(2n+1)x\bigr). \] The case $a=n!$ settles Sun's Conjecture 4.1.

math.NT↗

arXiv2025

Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem

We study the following perturbation from symmetry problem for the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~\Omega, \\[2mm] u\in H_{X,0}^{1}(\Omega),\hfill \end{array} \right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is the self-adjoint sub-elliptic operator associated with H\"{o}rmander vector fields $X=(X_{1},X_{2},\ldots,X_{m})$, $\Omega$ is an open bounded subset in $\mathbb{R}^n$, and $H_{X,0}^{1}(\Omega)$ denotes the weighted Sobolev space. We establish multiplicity results for sign-changing solutions using a perturbation method alongside refined techniques for invariant sets. The pivotal aspect lies in the estimation of the lower bounds of min-max values associated with sign-changing critical points. In this paper, we construct two distinct lower bounds of these min-max values. The first one is derived from the lower bound of Dirichlet eigenvalues of $-\triangle_{X}$, while the second one is based on the Morse-type estimates and Cwikel-Lieb-Rozenblum type inequality in degenerate cases. These lower bounds provide different sufficient conditions for multiplicity results, each with unique advantages and are not mutually inclusive, particularly in the general non-equiregular case. This novel observation suggests that in some sense, the situation for sub-elliptic equations would have essential difference from the classical elliptic framework.

math.AP↗

arXiv2025

Critical Sobolev inequalities and global extremals for homogeneous H\"{o}rmander vector fields on $\mathbb{R}^n$

We study critical Sobolev inequalities and Sobolev extremal functions for homogeneous H\"{o}rmander vector fields on $\mathbb R^n$. The focus is the non-equiregular case. In this setting, the sub-Riemannian flag may have different growth at different points, the volume of subunit balls is not governed by a single local dimension, and the translation structure available on Carnot groups is absent in general. These features make both the range of admissible Sobolev exponents and the attainment of the optimal Sobolev constant substantially more delicate, especially on unbounded domains. We first give a domain-dependent description of the admissible Sobolev exponents in terms of the volume growth rates of subunit balls. This description separates the roles of the pointwise homogeneous dimension, the non-isotropic dimension of the domain, and the singular strata that may be approached at infinity. As a consequence, we prove a global Sobolev inequality for all homogeneous H\"{o}rmander vector fields on $\mathbb R^n$, without assuming any underlying group law. We then prove that the optimal Sobolev constant is attained. The main new geometric ingredient is a smooth family of maps $T(w,x)$ along the maximal level set $H$ of the pointwise homogeneous dimension. These maps preserve Lebesgue measure and the horizontal gradient, and they play the role of left translations in a setting where no group multiplication is available. In this sense, $T(w,x)$ compensates for the lack of an intrinsic translation structure and allows one to recenter minimizing sequences in the critical concentration-compactness argument. The compactness theorem obtained in this way applies to all homogeneous H\"{o}rmander vector fields. We also prove that, on every open set meeting this maximal level set, the optimal Sobolev constant is independent of the domain.

math.AP↗

arXiv2024

Sharp embedding results and geometric inequalities for H\"{o}rmander vector fields

Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of H\"{o}rmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where $\Omega\subset\subset U$ is a general open bounded subset of $U$. By employing Rothschild-Stein's lifting technique and saturation method, we prove the representation formula for smooth functions with compact support in $\Omega$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where the critical Sobolev exponent depends on the generalized M\'{e}tivier index. As applications of these sharp Sobolev inequalities, we establish the isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and Moser-Trudinger inequality in the context of general H\"{o}rmander vector fields.

math.AP↗

arXiv2023

Multiplicity of solutions for semilinear subelliptic Dirichlet problem

In this paper, we study the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~\Omega, \\[2mm] u=0\hfill & \mbox{on}~\partial\Omega, \end{array} \right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is the self-adjoint H\"{o}rmander operator associated with vector fields $X=(X_{1},X_{2},\ldots,X_{m})$ satisfying the H\"{o}rmander condition, $f(x,u)\in C(\overline{\Omega}\times \mathbb{R})$, $g(x,u)$ is a Carath\'{e}odory function on $\Omega\times \mathbb{R}$, and $\Omega$ is an open bounded domain in $\mathbb{R}^n$ with smooth boundary. Combining the perturbation from symmetry method with the approaches involving eigenvalue estimate and Morse index in estimating the min-max values, we obtain two kinds of existence results for multiple weak solutions to the problem above. Furthermore, we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations. Compared with the classical elliptic cases, both approaches here have their own strengths in the degenerate cases. This new phenomenon implies the results in general degenerate cases would be quite different from the situations in classical elliptic cases.

math.AP↗

arXiv2022

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

This paper is concerned with existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.

math.AP↗

arXiv2022

Asymptotic behaviour of Dirichlet eigenvalues for homogeneous H\"{o}rmander operators and algebraic geometry approach

We study the Dirichlet eigenvalue problem of homogeneous H\"{o}rmander operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent smooth vector fields in $\mathbb{R}^n$ satisfying H\"{o}rmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that $\Omega$ is an open bounded domain in $\mathbb{R}^n$ containing the origin. We use the Dirichlet form to study heat semigroups and subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry, and employing some refined analysis involving convex geometry, we establish the explicit asymptotic behavior $\lambda_k \approx k^{\frac{2}{Q_0}}(\ln k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $\lambda_k$ denotes the $k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $\Omega$, $Q_0$ is a positive rational number, and $d_0$ is a non-negative integer. Furthermore, we provide optimal bounds of index $Q_0$, which depend on the homogeneous dimension associated with the vector fields $X_1, X_2, \ldots, X_m$.

math.AP↗