Degenerate conformally invariant fully nonlinear elliptic equations

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

实数域\(\mathbb{R}^n\)上具有非局部非线性的\(n\)阶共形不变椭圆方程解的分类

Classification of solutions to an $n^{\text{th}}$ order conformally invariant elliptic equation on $\mathbb R^n$ with nonlocal nonlinearity

研究实数域\(\mathbb{R}^n\)上由\((-\Delta)^{n/2}\)驱动、具非局部指数型Choquard非线性的共形不变椭圆问题，通过合适可积性假设对解在无穷处渐近行为分类，在能量下界等条件下给出解的明确分类。

The Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds

We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds under appropriate assumptions.

具有Choquard型非线性的共形不变椭圆方程的无穷多解

Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity

本文研究具有非局部临界幂非线性的共形不变椭圆方程，通过将问题提升到标准球面利用对称性克服Sobolev嵌入紧性失效，构造出两类无界变号解序列。

Escobar's Type Theorems for elliptic fully nonlinear degenerate equations

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are the hemisphere (or a geodesic ball in $\mathbb{S}^m$), punctured balls and annular domains. Our results extend those of Escobar in when $m\geq 3$, and Hang-Wang and Jimenez in when $m=2$.

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $\nabla^2 ψ+ L(x,ψ,\nabla ψ)$ which are non-decreasing in $ψ$.

The Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with mean concave boundary

This article studies the Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with \textit{mean concave} boundary in the sense that the mean curvature of the boundary is \textit{nonpositive}. The proof is primarily based on a quantitative boundary estimate. Also, we obtain analogous results in complex variables. In Appendix, the subsolutions are also constructed on certain topologically product manifolds.

Fully non-linear degenerate elliptic equations in complex geometry

We derive an a priori real Hessian estimate for solutions of a large family of geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent of a lower bound for the right-hand side function. This improves on the estimates of Székelyhidi and additionally applies to elliptic equations with a degenerate right-hand side. As an application, we establish the optimal $C^{1,1}$ regularity of envelopes of $(θ, m)$-subharmonic functions on compact Hermitian manifolds.

Boundary value problems for some fully nonlinear elliptic equations

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

庞加莱-爱因斯坦流形上退化与非局部椭圆方程的奇异集

The singular sets of degenerate and nonlocal elliptic equations on Poincaré-Einstein manifolds

本文针对庞加莱-爱因斯坦流形上自然出现的退化与非局部椭圆算子，建立定量分层和Minkowski型估计，并结合新的$ε$-正则性结果，得到奇异集的一致Hausdorff测度估计，揭示了几何与椭圆理论之间的精细协同。

Well-posedness of boundary value problems for a class of second order degenerate elliptic equations

In this paper, we study the well-posedness of boundary value problems for a special class of degenerate elliptic equations coming from geometry. Such problems is intimately tied to rigidity problem arising in infinitesimal isometric deformation, The characteristic form of this class of equations is changing its signs in the domain. Therefore the well-posedness of these above problems deserve to make a further discussion. Finally, we get the existence and uniqueness of $H^1$ solution for such boundary value problems.

Optimal gradient continuity for degenerate elliptic equations

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nabla u(X) = 0 \}$. The innovative feature of our main result concerns its optimality -- the sharp, encoded smoothness aftereffects of the operator. Such a quantitative information usually plays a decisive role in the analysis of a number of analytic and geometric problems. Our resu…

Min-oo conjecture for fully nonlinear conformally invariant equations

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$…

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…

A Liouville theorem of degenerate elliptic equation and its application

In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.

$W^{1,1}_0(Ω)$ in some borderline cases of elliptic equations with degenerate coercivity

We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.

A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type

We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.

Existence of multiple solutions for quasi-linear degenerate elliptic equations

The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity many solutions for the problems.

An Elliptic System with Degenerate Coercivity

In this paper we study the existence of solutions of thedegererate elliptic system.

A weak comparison principle for solutions of very degenerate elliptic equations

We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where $\nabla u \in K$, where $K\subset \mathbb{R}^N$ is a Borel set containing the origin.

Global Irregularity For Degenerate Elliptic Operators

Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.

具有变特征指数的一致退化椭圆方程

Uniformly Degenerate Elliptic Equations with Varying Characteristic Exponents

本文研究有界区域中特征多项式具变特征指数的一致退化椭圆方程，核心关注解的正则性问题。

关于广泛退化椭圆型系统的二阶正则性

On the second-order regularity of widely degenerate elliptic systems

研究广泛退化椭圆型系统二阶正则性，通过证明临界集附近正则性及退化区域测度性质得出相关结论。

完全非线性退化椭圆方程的Schauder型估计

Schauder-type estimates for fully nonlinear degenerate elliptic equations

该文针对完全非线性退化椭圆方程，研究其解在内部源项消失点处的正则性，得到了依赖类Hölder源项-椭圆性消失速率的Schauder型正则性估计。

Harnack inequality for degenerate and singular elliptic equations with unbounded drift

We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.

Continuity in two dimensions for a very degenerate elliptic equation

An elliptic equation div(F(Du)) = f whose ellipticity strongly degenerates for small values of Du (say, F = 0 on B(0,1)) is considered. The aim is to prove regularity for F(Du). The paper proves a continuity result in dimension two and presents some applications.

一类带临界增长和对数扰动的非线性退化椭圆算子的Dirichlet问题

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

该研究聚焦于一类带临界增长与对数扰动的非线性退化椭圆算子的Dirichlet问题，探究其弱解的存在性。

C^{1,α}-regularity for a class of degenerate/singular fully nonlinear elliptic equations

We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.

一致退化椭圆方程的最优边界正则性

Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations

本文综述了有界区域中一致退化椭圆方程解的最优正则性，建立了解及其导数直至边界的Hölder连续性。

The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity

In this article we study an elliptic problem with degenerate coercivity. We will show that the presence of some lower order terms has a regularizing effect on the solutions.

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.

Symmetry of singular solutions of degenerate quasilinear elliptic equations

We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.

Maximum and comparison principles for degenerate elliptic systems and some applications

In this paper we develop a detailed study on maximum and comparison principles for a degenerate elliptic system. Explicit lower bounds for principal eigenvalues of this system in terms of the measure of $Ω$ are also proved.

Well-posedness for degenerate elliptic PDE arising in optimal learning strategies

We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the differential operator to construct (logarithmically) diverging barriers.

The thin obstacle problem for some variable coefficient degenerate elliptic operators

In this paper we establish the optimal interior regularity and the $C^{1,γ}$ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.

Existence of solutions of degenerate semilinear elliptic boundary value problems

We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence result of large solution to a similar problem.

Semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$

This paper is concerned with semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of Lévy-Itô type. Hölder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by Richardson's method.

退化椭圆方程含低阶项解的弱最大原理

The weak maximum principle for solutions of degenerate elliptic equations with lower order terms

本文研究退化椭圆方程含低阶项解的弱最大原理，基于作者与他人最近证明的存在性结果提出新证明。

On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids

We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.

An eigenvalue problem involving a degenerate and singular elliptic operator

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.

Free boundary regularity for a degenerate fully non-linear elliptic problem with right hand side

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free boundaries are $C^{1, γ}$.

Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations

We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.

Regularity for degenerate two-phase free boundary elliptic problems

This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.

Calderón-Zygmund估计在A_p权重下退化椭圆PDE中失效

Failure of Calderón-Zygmund estimates for degenerate elliptic PDEs with $A_p$-weights when $p > 2$

本文通过凸积分技术展示，在A_p权重(p>2)下，退化线性椭圆PDE的加权Calderón-Zygmund估计失效。

各向异性退化椭圆方程解的二阶正则性

Second order regularity for solutions to anisotropic degenerate elliptic equations

本文研究退化各向异性椭圆方程解的正则性，建立了二阶估计并给出应力场正则性结果，且结果在欧几里得情形下亦为新。

Strong Comparison Principles for Some Nonlinear Degenerate Elliptic Equations

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form $\nabla^2 ψ+ L(x,\nabla ψ)$, including the conformal hessian operator.

Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.

Weighted $W^{1,p}$- estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable

This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The coefficient matrix $\mathbf{A}(X)$ is a symmetric, measurable $(n+1) \times (n+1)$ matrix, and it could be degenerate or singular in the one dimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ of weights. Our results give weighted Sobolev regularity estimates of Calderón-Zygmund type for weak solutions of this class of singular, degene…

A Dirichlet problem for nonlocal degenerate elliptic operators with internal nonlinearity

We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.

穿孔区域中退化椭圆方程的一致正则性

Uniform regularity for degenerate elliptic equations in perforated domains

针对穿孔区域中快速振荡系数的退化椭圆方程，建立了定量收敛速率，并获得了加权的Lipschitz和$W^{1,p}$一致估计。

An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term

In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions

Entire subsolutions of fully nonlinear degenerate elliptic equations

We prove existence and non existence results for fully nonlinear degenerate elliptic inequalities, by showing that the classical Keller--Osserman condition on the zero order term is a necessary and sufficient condition for the existence of entire sub solutions.

Optimal C^{1,\apha} regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition

In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.