arXiv subjects

Chunqin Zhou

Publications and source records attributed to Chunqin Zhou.

14 recordsLinked to original sources

arXiv2026

Boundary pointwise regularity for the Poisson problem on uniform domain

In this paper, we study the boundary pointwise regularity for the Poisson problem on domains with rough boundaries, specifically uniform domains. In general, it is not straightforward to define weak solutions for non-zero boundary data on such domains. To address this, we introduce a novel definition of weak solutions tailored to the setting of uniform domains. Remarkably, this definition allows for the analysis of the regularity of weak solutions. In particular, by establishing an energy inequality, we prove the boundary pointwise $C^α$ regularity by using compactness methods under the admissible condition. Furthermore, by exploiting the the linear structure of solutions with respective to the harmonic functions, we establish boundary pointwise $C^{1,α}$ and $C^{2,α}$ regularities when the boundary data and the domain boundary are pointwise $C^{1,α}$ and $C^{2,α}$, respectively.

math.AP↗

arXiv2026

On the super-Liouville equations on the sphere

In this paper, we investigate the existence of nontrivial least-energy solutions for the super-Liouville equation with positive coefficient functions on the two-dimensional sphere. Firstly, we derive a global Pohozaev-type identity by analyzing the behavior of solutions under conformal transformations, which generalizes the classical Kazdan-Warner obstruction for the two-dimensional Nirenberg problem. Secondly, by exploiting conformal symmetry, we establish a pointwise estimate that bounds the norm of the spinor component by the scalar component, and show that the $H^1 \times H^{1/2}$ energy of the spinor part remains uniformly bounded. As a byproduct of our analysis, parallel techniques are applied to the Dirac-Einstein equations on the 3-sphere, demonstrating that nontrivial solutions are uniformly bounded away from the trivial solution in the $H^1 \times H^{1/2}$ norm. Moreover, the compactness of the solution space is also analyzed from two perspectives: in the low-energy regime, and modulo the action of the Möbius group. Finally, by introducing a new natural constraint $\mathcal{A}$ and employing variational methods, we obtain a supersymmetric generalization of the Moser-Trudinger-Onofri inequality and establish the existence of least-energy solutions for even coefficient functions. In particular, these solutions are shown to be nontrivial provided that a certain spectral parameter associated with the coefficients satisfies $λ_1(h_2, h_1) < 1$. Concurrently, we provide a complete classification of nontrivial least-energy solutions in the case of positive constant coefficients.

math.AP↗

arXiv2025

The singular anisotropic Adams' type inequality in $\mathbb{R}^n$

In this paper, using anisotropic rearrangement techniques, we first establish the best constants for the singular anisotropic Adams' type inequality with exact growth in $\mathbb{R}^n$. Furthermore, by the same trick, we also prove the singular anisotropic Adams' type inequality on bounded domain $Ω\subset \mathbb{R}^n$.

math.AP↗

arXiv2024

Existence of Solutions to a super-Liouville equation with Boundary Conditions

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemannian surface $M$ with boundary and with its Euler characteristic $χ(M)<0$. The boundary condition couples a Neumann condition for functions and a chirality boundary condition for spinors. Due to the generality of the equation, we introduce a weighted Dirac operator based on the solution to a related Liouville equation. Then we construct a Nehari manifold according to the spectral decomposition of the weighted Dirac operator, and use minimax theory on this Nehari manifold to show the existence of the non-trivial solutions.

math.AP↗

arXiv2023

$C^{0}$-regularity for solutions of elliptic equations with distributional coefficients

In this paper, the continuity of solutions for elliptic equations in divergence form with distributional coefficients is considered. Inspired by the discussion on necessary and sufficient conditions for the form boundedness of elliptic operators by Maz'ya and Verbitsky (Acta Math., 188, 263-302, 2002 and Comm. Pure Appl. Math., 59, 1286-1329, 2006), we propose two kinds of sufficient conditions, which are some Dini decay conditions and some integrable conditions named Kato class or $K^{1}$ class, to show that the weak solution of the Schrödinger type elliptic equation with distributional coefficients is continuous and give an almost optimal priori estimate. These estimates can clearly show that how the coefficients and nonhomogeneous terms influence the regularity of solutions. The $\ln$-Lipschitz regularity and Hölder regularity are also obtained as corollaries which cover the classical De Giorgi's Hölder estimates.

math.AP↗

arXiv2022

Boundary Lipschitz regularity of solutions for general semilinear elliptic equations in divergence form

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.

math.AP↗

arXiv2021

Boundary Lipschitz regularity of solutions for semilinear elliptic equations in divergence form

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.

math.AP↗

arXiv2019

Energy quantization for a singular super-Liouville boundary value problem

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

math.DG↗

arXiv2017

Vanishing Pohozaev constant and removability of singularities

Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity. In situations where such an identity fails to hold, we introduce a new quantity, called the {\it Pohozaev constant}, which on one hand measures the extent to which the Pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities, conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing.

math.DG↗

arXiv2017

The Super-Toda System and Bubbling of Spinors

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy identities for the spinor parts of a blow-up sequence of solutions for which there are possibly four types of bubbling solutions, namely, finite energy solutions of the super-Liouville equation or the super-Toda system defined on $\R^2$ or on $\R^2\setminus\{0\}$. This is achieved by showing some new energy gap results for the spinor parts of these four types of bubbling solutions.

math.DG↗

arXiv2015

Towards Grothendieck Constants and LHV Models in Quantum Mechanics

We adopt a continuous model to estimate the Grothendieck constants. An analytical formula to compute the lower bounds of Grothendieck constants has been explicitly derived for arbitrary orders, which improves previous bounds. Moreover, our lower bound of the Grothendieck constant of order three gives a refined bound of the threshold value for the nonlocality of the two-qubit Werner states.

quant-ph↗

arXiv2012

Local Unitary Equivalence of Arbitrary Dimensional Bipartite Mixed Quantum States

The nonlocal properties of arbitrary dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. These invariants give rise to both sufficient and necessary conditions for the equivalence of quantum states under local unitary transformations: two density matrices are locally equivalent if and only if all these invariants have equal values.

quant-ph↗

arXiv2007

Metrics of constant curvature on a Riemann surface with two corners on the boundary

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curvature and constant geodesic curvature on its boundary if and only if the two corners have the same angle. In fact, we can classify all the solutions in a more general situation, that of the 2-sphere cut by two planes.

math.DG↗

arXiv2005

Super-Liouville Equations on Closed Riemann Surfaces

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.

math.DG↗