We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying geometric principle, Theorem \ref{theorem KM}, is classical and well known \cite[Theorem 1]{KM}; it gives necessary and sufficient conditions for a connection to induce a holomorphic structure on a vector bundle over a complex manifold. Here we explore the limits of this statemen…

2013-09-18math.APmath.DG

Remarks on extremals of sharp Sobolev trace inequalities on the unit balls

We show explicit forms for extremals of some fourth-order sharp trace inequalities on the unit balls recently proved by Ache-Chang. We also give a classification result of the bi-harmonic equation on $\mathbb{R}^4_+$ with some conformally covariant boundary conditions.

2022-10-01math.AP

Conservation laws for conformal invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting…

2006-03-15math.APmath.DG

A remark on observability of the wave equation with moving boundary

We deal with the wave equation with assigned moving boundary ($0<x<a(t)$) upon which Dirichlet or mixed boundary conditions are specified, here $a(t)$ is assumed to move slower than the light and periodically. Moreover $a$ is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each $t>0$ of the transverse velocity at $a(t)$. The key to the results is the use of a reduction theorem.

2015-07-20math.AP

A Remark on Gauge Transformations and the Moving Frame Method

In this note we give a shorter proof of recent regularity results by Riviere and Riviere-Struwe. We differ from the mentioned articles only in using the direct method of Helein's moving frame to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using besides the duality of Hardy- and BMO-space only elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandt's conjecture one can avoid the Nash-Moser imbedding theorem. There are no new results pr…

2009-06-10math.AP

三维双曲空间上具有负临界指数的共形不变方程

Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space

Indian Statistical Institute（印度统计研究所）；University of Science and Technology of China（中国科学技术大学）；University of Connecticut（康涅狄格大学）；Rutgers University（罗格斯大学）

研究三维双曲空间上四阶方程正解的增长率对称性，证明正解必须径向严格递减。

2026-03-13math.AP

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关于运动区域中线性化微极流体系统分层控制的注记

Remarks on Hierarchic Control for a Linearized Micropolar Fluids System in Moving Domains

本文研究运动边界区域内线性化微极流体方程的分层Stackelberg控制策略，推导纳什均衡的存在唯一性与刻画、系统对领导者控制的近似能控性，以及Stackelberg-Nash问题的适定性与领导者最优性系统。

2025-01-11math.AP

共形不变完全非线性方程的Liouville定理. I

Liouville theorems for conformally invariant fully nonlinear equations. I

本文研究共形不变完全非线性椭圆方程的Liouville定理，给出其成立及局部梯度估计的充要条件，并应用于闭流形上共形度量的存在性与紧致性。

2024-09-19math.APmath.DG

An extension problem for the CR fractional Laplacian

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

2013-12-12math.APmath.DG

On a family of integral operators on the ball

In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

2021-10-18math.APmath.DG

关于构造双调和与共形双调和映射到球面的remarks

Remarks on constructing biharmonic and conformal biharmonic maps to spheres

University of Rostock（罗斯托克大学）

本文提出一种几何算法，用于构造双调和与共形双调和映射到球面，并通过实例展示算法在不同域下的灵活性及方法的有效性。

2026-03-06math.APmath.DG

Classification of solutions of an equation related to a conformal log Sobolev inequality

We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from $\mathbb R^n$ to $\mathbb S^n$ and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.

2020-03-18math.AP

Remarks on space-time behavior in the Cauchy problems of the heat equation and the curvature flow equation with mildly oscillating initial values

We study two initial value problems of the linear diffusion equation and a nonlinear diffusion equation, when Cauchy data are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the moving curves are represented by graphs. By using an elementary scaling technique, we show some formulas for space-time behavior of the solution. Keywords: scaling argument, self-similar solution, nonstabilizing solution, nontrivial dynamics, nontrivial large-time behavior, irregular behavior.

2012-03-20math.AP

Conformal flow on $S^3$ and weak field integrability in AdS$_4$

We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the …

2017-03-23gr-qchep-thmath-ph

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $γ$-gas law equation of state for $γ=2$ and the general initial density $\ri \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the …

2013-06-20math.AP

Classification of solutions for some mixed order elliptic system

In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ \mathbb{R}^4$: \begin{equation}\left\{ \begin{aligned} & -Δu(x) = u^{p_1}(x) e^{q_1v(x)}, \quad x\in \mathbb{R}^4,\\ & (-Δ)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, \quad x\in \mathbb{R}^4, \end{aligned} \right. \end{equation} where $ 0\leq p_1 < 1$, $ p_2 >0$, $ q_1 > 0$, $ q_2 \geq 0$, $ u>0$ and satisfies $$ \int_{\mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx < \infty,\quad \int_{\mathbb{R}^4} u^{p_2}(x) e^{q_2 v(x)} dx < \infty.$$ Under additional assumptions (H1) or (H2), we …

2022-11-25math.AP

当内在增长率在有限大小的移动斑块中最大时 Fisher-KPP 方程的传播性质

Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

本文研究一维时空异质 Fisher-KPP 方程在中央快速生长斑块移动时的传播性质，揭示了非局部牵引和锁定等现象，并在斑块恒速恒大小情形下给出了传播速度的完整显式公式。

2025-11-06math.AP