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Chilin Zhang

Publications and source records attributed to Chilin Zhang.

18 recordsLinked to original sources
arXiv2026

Flexibility and rigidity of steady states of the two-dimensional Euler equations in an infinite channel

We study steady solutions to the two-dimensional incompressible Euler equations in an infinite channel, whose far-field limits are uniformly non-stagnant shear flows. In the smooth category,for a broad class of prescribed far-field shear profiles, non-shear steady states exist via the construction of two-dimensional solutions of the semilinear elliptic equations of stream function by the min--max method. In the analytic category, we establish a comparison principle for the analytic steady states and we show for a dense family of analytic uniformly non-stagnant shear profiles, every analytic steady state with the prescribed far field must itself be a shear flow. In particular, there are far-field shear profiles which exhibit flexibility in the smooth category but rigidity in the analytic category. Furthermore, the dense rigidity is sharp in the sense that there exists analytic shear profile which admits flexibility in the analytic category.

math.AP↗
arXiv2026

Classification of global solutions to the singular equation $-\Delta u=f(X)\cdot u^{-\gamma}$ in a Lipschitz epigraphical cone

We construct and classify all global solutions to the singular equation $$-\Delta u=f(X)\cdot u^{-\gamma}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<\lambda\leq f(X)\leq\Lambda$. The existence and non-existence of a global solution is solely determined by the exponent $\gamma$ of the equation and the ``frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the ``asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an ``oscillation reduction" argument on the ``asymptotic slope".

math.AP↗
arXiv2026

Curvature estimate for the heteroclinical solution to a Bose-Einstein condensation system

We investigate heteroclinical solutions of a vector-valued Bose-Einstein condensation system involving the $p$-Laplacian. The main difficulty comes from the degeneracy of the $p$-Laplacian and the possible nonsmooth behavior of the potential wells. Under suitable assumptions on the double-well potential, we establish the existence and detailed asymptotic behavior of heteroclinical solutions. In particular, we prove the strict monotonicity of every component, classify the blow-up profiles near the potential wells through Weiss-type monotonicity formulas, and obtain an almost homogeneity property of the solution. Furthermore, after re-parametrizing the heteroclinical solution by its $l_p$-arc length, we derive a curvature estimate for the trajectory near the potential wells. These results provide the geometric control needed for the construction of radial barrier functions for the corresponding Bose-Einstein condensation system.

math.AP↗
arXiv2026

Pointwise arbitrarily high-order interior estimates for mixed local and nonlocal elliptic equations

In this paper, we mainly focus on pointwise, arbitrarily high-order interior estimates for the mixed local-nonlocal elliptic equation \begin{equation*} (-\Delta)^{s}u(x)-\Delta u(x)=f(x),\quad x\in B_r(0) \end{equation*} with $0<s<1$. The main challenges in this setting are the absence of an explicit Green function and the ineffectiveness of standard bootstrap arguments. Our approach overcomes these difficulties via the Campanato iteration method, which inductively constructs polynomials approximating the solution. Using the fact that all functions are locally $s$-harmonic up to a small error significantly reduces the computational complexity when studying the regularity of the fractional Laplacian acting on these polynomials. Finally, the regularity estimates are obtained from the analysis of a manageable recursive inequality system.

math.AP↗
arXiv2026

Classification of solutions to a weighted singular fractional problem in the half space

We focus on the classification of positive solutions to $(-\Delta)^s u=\frac{x_n^{\alpha}}{u^\gamma}$ in the half space with $\gamma>0$, subject to the Dirichlet condition. We show that when $-2s<\alpha<(\gamma-1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope". When $\alpha$ lies outside this range, we demonstrate the nonexistence of global positive solutions.

math.AP↗
arXiv2026

Green function rigidity for two dimensional sphere

We verify a conjecture proposed by X. Chen and Y. Shi, which arises from their study of the Green function on spheres in Euclidean space. More precisely, let $M\subset \mathbb{R}^3$ be a closed $C^{2}$ embedded surface and suppose that there exists a point $p\in M$ so that its Green function $G$ is of the form $G(p,q)=-\frac{1}{2\pi} \ln d_{\mathbb{R}^3}(p,q)+c, \forall q\neq p$, then $M$ must be a round sphere.

math.DG↗
arXiv2025

Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$

In this series of papers, we investigate coupled systems arising in the study of two-component Bose-Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the present (second) paper of the series, we focus on the nonlocal problem of the form \begin{equation*} \left\{\begin{aligned} (-\Delta)^{s}u+u(u^{2}+v^{2}-1)+v(\alpha uv-\omega)=0, (-\Delta)^{s}v+v(u^{2}+v^{2}-1)+u(\alpha uv-\omega)=0, \end{aligned} \right. \end{equation*} which models the stationary states of Rabi-coupled condensates with inter- and intra-species interactions. We prove that for $\frac{1}{2}\le s<1$, any positive entire solution $(u,v)$ in $\mathbb{R}^3$ satisfying the monotonicity condition $\partial_{x_3}u>0>\partial_{x_3}v$ must be one-dimensional. Moreover, when $0<s<\frac{1}{2}$, the same conclusion holds for monotone solutions in $\mathbb{R}^2$. Our work generalizes classical De Giorgi-type theorems to a new class of nonlocal coupled systems and, to the best of our knowledge, presents the first Liouville-type classification of monotone solutions for Rabi-coupled fractional Bose-Einstein condensates, with particular emphasis on fractional Gross-Pitaevskii models.

math.AP↗
arXiv2025

Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in $\mathbb{R}^{3}$

In this series of papers, we investigate coupled systems arising in the study of two-component Bose--Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the first paper of the series, we focus on the local problem of the form $\Delta u = u(u^2+v^2-1) + v(\alpha uv - \omega)$, $\Delta v = v(u^2+v^2-1) + u(\alpha uv - \omega)$, and prove that positive global solutions in $\mathbb{R}^3$ satisfying $\partial u/\partial x_3 > 0 > \partial v/\partial x_3$ must be one-dimensional.

math.AP↗
arXiv2025

A further remark on the density estimate for degenerate Allen-Cahn equations: $\Delta_{p}$-type equations for $1<p<\frac{n}{n-1}$ with rough coefficients

In this short remark on a previous paper \cite{SZ25}, we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies \begin{equation*} J(v,\Omega)=\int_{\Omega}\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*} involving a Dirichlet energy $F(\vec{\xi},\tau,x)\sim|\vec{\xi}|^{p}$ and a degenerate double-well potential $W(\tau,x)\sim(1-\tau^{2})^{m}$. In contrast to \cite{SZ25}, we remove all regularity assumptions on the Ginzburg-Landau energy. Then, with further assumptions that $1<p<\frac{n}{n-1}$ and that $W(\tau,x)$ is monotone in $\tau$ on both sides of $0$, we establish a density estimate for the level sets of nontrivial minimizers $|u|\leq1$.

math.AP↗
arXiv2025

Sharp Boundary Growth Rate Estimate of the Singular Equation $-\Delta u=u^{-\gamma}$ in a Critical Cone

For $\gamma>0$, we study the sharp boundary growth rate estimate of solutions to the Dirichlet problem of the singular Lane-Emden-Fowler equation \begin{equation*} -\Delta u=u^{-\gamma} \end{equation*} in a critical $C^{1,1}$ epigraphical cone $Cone_{\Sigma}$. We show that the growth rate estimate exhibits fundamentally different behaviors in the following three cases: $1<\gamma<2$, $\gamma=2$, and $\gamma>2$. Moreover, we obtain the sharp growth rate estimate near the origin for $\gamma>1$. As a consequence, we show that when $Cone_{\Sigma}$ is a $C^{1,1}$ epigraphical cone, the additional solvability condition in \cite[Theorem 1.3]{GuLiZh25} is both sufficient and necessary to achieve the growth rate therein, thereby resolving the main open question left in that paper. With the growth rate estimate, we also derive the optimal modulus of continuity for solutions via the interior Schauder estimate. Our approach is to control the values of a solution $U(x)$ in the region $\Omega=Cone_{\Sigma}\cap B_{1}$ by introducing a sequence of reference points $p_{k}=\frac{16^{1-k}}{2}\vec{e_{n}}$. From the Green function representation of $U(x)$, we derive a discrete integral equation for the sequence $a_{k}=16^{k\phi}U(p_{k})$. Such a computation converts the original PDE problem into a recursion for a discrete integral equation, which can be effectively analyzed using basic ODE methods.

math.AP↗
arXiv2025

Density estimates for Ginzburg-Landau energies with degenerate double-well potentials

We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \begin{equation} J(v,\Omega)=\int_{\Omega}\Big\{|\nabla v|^{p}+(1-v^{2})^{m}\Big\}dx,\quad 1<p<m, \end{equation} and establish density estimates for the level sets of nontrivial minimizers $|v| \leq 1$. This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. The original estimates for the classical case $p=m=2$ were established by Caffarelli-C\'ordoba.

math.AP↗
arXiv2025

Boundary regularity theory of the singular Lane-Emden-Fowler equation in a Lipschitz domain

We study the singular Lane-Emden-Fowler equation \begin{equation} -\Delta u=f(X)\cdot u^{-\gamma} \end{equation} in a bounded Lipschitz domain $\Omega$, with the Dirichlet boundary condition and a positive, bounded function $f(X)$. A distinguishing feature is that the vanishing boundary condition introduces a singularity in the equation. We focus on the well-posedness of the equation and the growth rate of solutions near the boundary. The key is to classify the limiting cone of a boundary point into three categories based on its "frequency", and obtain distinct growth rate estimates for each case. Additionally, we discuss the boundary Harnack principle for the singular Lane-Emden-Fowler equation, which is essential in deriving the boundary growth rate estimate. To our knowledge, the boundary Harnack principle we derive is the first Kemper-type estimate for singular semi-linear equations. It notably differs from the classical one for linear equations, in particular, the boundedness of the ratio \(u/v\) does not imply its continuity. To address the lack of a suitable upper barrier, we introduce new techniques, including constructing upper barriers iteratively. We also construct a subharmonic auxiliary function $V(X)$ related to the solution $u$ in the limiting cone. The growth rate of $u(X)$ is then obtained inductively from the growth rate of the auxiliary function $V(X)$. Our results and methods offer novel insights into the behavior of singular elliptic equations in non-smooth domains.

math.AP↗
arXiv2025

Minimizers of the Allen-Cahn energy with sub-quadratic growth

We establish Liouville theorems for global minimizers $u$ of the Allen-Cahn energy $$\int |\nabla u|^2 + W(u) \, dx,$$ which have subquadratic growth at infinity. In particular we extend the results of \cite{S1,S3} concerning the De Giorgi's conjecture to the setting of unbounded solutions. Part of the analysis relies on the regularity of minimizers for a Dirichlet/perimeter functional which was studied by Athanasopoulous-Caffarelli-Kenig-Salsa in \cite{ACKS}.

math.AP↗
arXiv2024

An area growth estimate of the Liouville equation

We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $\Delta u+K e^{2u}=0$ with a positive pinched curvature $0<\lambda\leq K\leq\Lambda$. As an application, we provide a new proof of Eremenko-Gui-Li-Xu's result in [EGLX]. We also classify solutions with an upper bound in the half plane with the boundary having constant geodesic curvature.

math.AP↗
arXiv2023

Optimal H{\"o}lder convergence of a class of singular steady states to the Bahouri-Chemin patch

Singular steady states are important objects in obtaining ill-posedness results for 2D incompressible Euler equations. In \cite{elgindi2022regular}, a family of singular steady states near the Bahouri-Chemin patch was introduced. In this paper, we obtain the optimal convergence results for the singular steady states constructed in \cite{elgindi2022regular} to the Bahouri-Chemin patch. We first derive a boundary Harnack principle, and then obtain the optimal convergence results using the singular integral representation based on Green's function.

math.AP↗
arXiv2023

On higher order boundary Harnack and analyticity of free boundaries

We establish a C^1,alpha Schauder estimate of a non-standard degenerate elliptic equation and use it to give another proof of the higher order boundary Harnack inequality. As an application, we obtain the analyticity of the free boundary in the classical obstacle problem based on iterating the boundary Harnack principle.

math.AP↗