Yanyan Li
This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
We study solutions to conformally invariant equations with isolated singularties.
We establish Liouville type theorems for degenerate conformally invariant equations.
In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.
We establish a Liouville type theorem for some conformally invariant fully nonlinear equations
We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.
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 $\partial D(r)$…
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。
We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.
We solve the $σ_2$-Yamabe problem for a non locally conformally flat manifold of dimension $n>8$.
We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.
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.
On the fully nonlinear Yamabe problem with constant boundary mean curvature. I
本文研究了完全非线性Yamabe问题，证明了共形不变二阶椭圆方程在半欧氏空间中的最优Liouville型定理。
Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.
We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone $Γ$ satisfies $μ^+_Γ\le 1$, which includes the $σ_k-$Yamabe problem for $k$ not smaller than half of the dimension of the manifold.
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant condition for positivity of the Paneitz operator, which allows us to give many new examples of manifolds admitting metrics with constant $Q$-curvature.
We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.
In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class of even-dimensional complete LCF manifolds with integrable $Q$% -curvature, we establish a Gauss-Bonnet-Chern inequality. Second, a finiteness theorem for certain classe…
One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homoth…
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Gauduchon which extends the Calabi conjecture; this was one of the original motivations to this work. We were also motivated by the fact that there had been increasing interests in fully nonlinear pde's from complex geometry in recent years, and aimed to develop general methods to cover as wide a class of equations as possible.
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Ampère equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Ampère equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercrit…
Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds
本文针对紧致厄米流形上的凹完全非线性椭圆方程，提出一种不依赖Hou-Ma-Wu估计和Liouville定理的梯度估计直接证明方法，并结合稳定性估计与比较论证，同时给出紧致Kähler流形上复Hessian估计的独立证明。
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 $ψ$.
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for admissible solutions, then we obtain boundary gradient estimates based on the interior gradient estimates we have got.
Gradient regularity for fully nonlinear equations with degenerate coefficients
针对在超曲面上退化的完全非线性方程，推导了粘性解的$C^{1,α}$正则性估计。
In this paper we shall establish some regularity results of solutions of a class of fully nonlinear equations, with a first order term which is sub-linear. We prove local Hölder regularity of the gradient both in the interior and up to the boundary.
Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
本文针对完全非线性退化或奇异抛物型方程，利用紧性方法、障碍函数构造、正则化技术及小扰动解的边界正则性，建立了统一的边界Hölder梯度估计，推广了已有结果。
In this paper we consider second order fully nonlinear operators with an additive superlinear gradient term. Like in the pioneering paper of Brezis for the semilinear case, we obtain the existence of entire viscosity solutions, defined in all the space, without assuming global bounds. A uniqueness result is also obtained for special gradient terms, subject to a convexity/concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case.
Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results
本文引入全非线性方程的P-函数概念，建立一般准则，获得梯度界、De Giorgi型性质与刚性结果，推广Modica不等式，并证明Harnack型不等式及高阶方程的Liouville定理。
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_{t}=F(D^2u,D u,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applications are given, including the standard porous medium equation.
Gradient regularity for degenerate fully nonlinear free transmission problems with Hamiltonian terms
研究含哈密顿项的退化完全非线性自由传输问题，通过置于粘性不等式背景建立梯度局部赫尔德正则性，基于新估计与局部分析获逐点\(C^{1,\alpha}\)正则性。
Sharp gradient estimates for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions and Hamiltonian terms
研究一类具斜边界条件和哈密顿项的奇异/退化完全非线性椭圆方程，在一般条件下，于统一框架内建立直至边界的精确 \(C^{1,\alpha}\) 正则性。
We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the sec…
Interior Hölder and Calderón-Zygmund estimates for fully nonlinear equations with natural gradient growth
本文针对梯度二次增长且右端无界的完全非线性二阶方程，通过非线性 Harnack 不等式建立了粘性解的局部 Hölder 估计，并导出了新的 Liouville 性质与非线性 Calderón-Zygmund 估计。
Gradient ascent method for fully nonlinear parabolic differential equations with convex nonlinearity
该研究针对含凸非线性项的完全非线性抛物型PDE，提出将其转化为半线性问题并利用扩散系数方向最大值原理的梯度上升数值法，通过投资组合优化的HJB方程实验验证了可行性。
We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent, earlier ones via a totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear problems in t…
We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known a…
In this paper we prove Holder regularity of the derivative of radial solutions to fully nonlinear equations when the operator is hessian, homogenous of degree 1 in the Hessian, homogenous of some degree $α>-1$ in the gradient and which is elliptic when the gradient is not null.
A new proof of Hölder estimates for the gradient of quasilinear elliptic equations
本文受Evans-Krylov定理证明启发，采用覆盖方法给出拟线性椭圆方程梯度Hölder估计的新证明，并通过相同方法获得全非线性椭圆方程梯度的Hölder估计。
In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi-Mingione. We however mention that, differently from the approach in the interior case which depends on $W^{1,q}$ estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental work…
Regularity for Fully Nonlinear Elliptic Equations with Natural Growth in Gradient and Singular Nonlinearity
本文研究梯度超线性增长且含奇异非线性项u^(-α)的完全非线性椭圆方程，在系数c(x)满足负上界条件下，分析解的边界行为并建立全局正则性。
We propose a least squares Galerkin based gradient recovery to approximate Dirichlet problems for strong solutions of linear elliptic problems in nondivergence form and corresponding apriori and aposteriori error bounds. This approach is used to tackle fully nonlinear elliptic problems, e.g., Monge-Ampère, Hamilton-Jacobi-Bellman, using the smooth (vanilla) and the semismooth Newton linearization. We discuss numerical results, including adaptive methods based on the aposteriori error indicators.
The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of proper elliptic branches inspired by Krylov (Trans. Amer. Math. Soc. 1995) with the monotonicity-duality method initiated by Harvey and Lawson (Comm. Pure Appl. Math. 2009). In the variable coefficient nonlinear potential theory, a special role is played by the Hausdorff continuity of the prope…
Gradient Regularity for Fully Nonlinear Equations with Variable Degeneracy and Hamiltonian Lower-Order Terms
研究具有可变梯度退化和Hamilton型低阶项的全非线性椭圆方程的粘性解，通过紧致性估计和扁平化改进迭代，证明了内梯度Hölder估计、源项和Hamilton系数消失点的逐点改进以及极值点的Schauder型估计。
Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets
该研究针对仅在解梯度位于闭球和闭纯一维不可求集外时成立的完全非线性一致椭圆方程，证明了与例外集无关的粘性解的哈纳克不等式和赫尔德正则性估计，还通过例子和反例说明纯一维不可求性的作用，且内容自包含。
We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as $$ -F(x,u,Du,D^2u) =λc(x)u+\langle M(x)D u, D u \rangle +h(x) $$ in a bounded domain with a Dirichlet boundary condition, here $λ\in\mathbb{R}$, $c,\, h \in L^p(Ω)$, $p>n\geq 1$, $c\gneqq 0$ and the matrix $M$ satisfies $0<μ_1 I\leq M\leq μ_2 I$. Recently this problem was studied in the "coercive" case $λc\le0$, where uniqueness of solutions can be expected, and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya,…
We prove new borderline regularity results for solutions to fully nonlinear elliptic equations together with pointwise gradient potential estimates.
Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere
本文研究球面上 Chang-Gui 型 Trudinger-Moser-Onofri 不等式的定量稳定性，利用泛函的共形不变性刻画全部极值函数，并在梯度范数下建立尖锐稳定性估计。
Multiplicity results for fully nonlinear elliptic equations with natural gradient growth
本文研究带自然梯度增长的全非线性椭圆方程边值问题，通过构造两对有序上下解，证明了对一类次线性增长的 f 该问题存在三个正解。
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi an…
We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties unconditionally for general non-linear diffusions and interaction potentials, while the second-order scheme does so provided a CFL condition holds. Sweeping dimensional splitting permits the effic…
We consider the problem of finding $λ\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{λ+ F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic, positively homogeneous and superadditive, $f$ is convex and superlinear, and $H$ is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique $λ^*$ for which the above equation has a solution $u$ with appropriate growth as $|x|\rightarrow \infty$. Moreover, associated to $λ^*$ is a convex so…
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations u…
Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations
本文研究一类在解梯度为零时退化的完全非线性积分微分方程，利用紧致性和扰动论证，根据不同的分数阶扩散阶数完整刻画了粘性解的正则性。
Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
本文针对 Gross-Pitaevskii 特征值问题基态计算，提出基于 $H^1$ 范数的全离散 Sobolev 梯度流方法，证明了全局收敛性、局部指数收敛性及 $P^1$ 有限元下与网格无关的收敛速率，并通过二维和三维高阶谱元数值实验验证了方法有效性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。