Szu-yu Sophie Chen
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.
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We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.
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.
The paper concerns singular solutions of nonlinear elliptic equations.
We obtain local pointwise second derivative estimates for $W^{2,p}$-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.
We study solutions to conformally invariant equations with isolated singularties.
We establish Liouville type theorems for degenerate conformally invariant equations.
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.
We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.
Fully nonlinear elliptic equations for some prescribed curvature problems on Hermitian manifolds
该研究利用爆破论证与部分一致椭圆性研究Hermitian流形上的完全非线性椭圆方程，将其应用于寻找指定Chern-Ricci曲率的共形Hermitian度量的几何问题，且假设因几何函数理论阻碍几乎达最优。
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.
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.
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 $m=2$.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.
We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new proof of an estimate of Streets-Warren on the twisted real Monge-Ampere equation.
In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge--Ampère equation for $(n-1)$-plurisubharmonic equations in the almost Hermitian setting.
Note on Regularity theory for Nonlinear elliptic equations
本文综述了非线性椭圆方程正则性理论的重要发展，涵盖De Giorgi-Nash-Moser理论及Krylov-Safonov和Evans-Safonov理论的核心贡献。
We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.
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.
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.
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary except being smooth and compact. As applications of these estimates we obtain results on regularity and existence.
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.
Solvability of a class of fully nonlinear elliptic equations on tori
该研究聚焦平坦环面上一类完全非线性椭圆方程的可解性，这类方程源自环面丛相关Calabi-Yau型问题的研究，为相关几何分析问题提供支撑。
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.
We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equati…
Fully non-linear elliptic equations on noncompact complex manifolds
本文研究非紧复流形上非线性椭圆方程的先验估计与存在性，应用包括构造具有给定体积形式的凯勒度量及爱因斯坦度量。
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 $ψ$.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.
We consider a class of fully nonlinear second order elliptic equations on Hermitian manifolds closely related to the general notion of $\bfG$-plurisubharmonicity of Harvey-Lawson and an equation treated by Székelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions we derive interior estimates and establish the existence of smooth solutions for the Dirichlet problem as well as for equations on closed manifolds.
In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$Δu+cu^α=0,$$ where $c, α$ are two real constants and $c\neq 0$.
This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree distinct solutions.
In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.
A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.
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