相关论文: Interior $C^2$ estimate for Hessian quotient equat…
The curvature estimates of $k$ curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the $n-1$ case and we also discuss some applications for our estimate.
In this paper, we establish global C^2 estimates for a class of mixed Hessian equations with Neumann boundary condition, and obtain the existence theorem of k-admissible solutions for the classical Neumann problem of these mixed Hessian…
We consider the Hamiltonian stationary equation for all phases in dimension two. We show that solutions that are $C^{1,1}$ will be smooth and we also derive a $C^{2,\alpha}$ estimate for it.
We establish a prior interior $C^{1,1}$ estimates for convex solutions and supercritical phase solutions to the Lagrangian mean curvature equation with sharp Lipschitz phase. Counter-examples exist when the phase is H\"{o}lder continuous…
Global second order H\"{o}lder regularity for Stokes systems can be obtained by global Schauder estimates, which are actually a priori estimates and were established by Solonnikov [20] and [23] with appropriate compatible conditions. This…
We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized…
We develop an integral approach to obtain interior a priori $C^{1,1}$ estimates for convex solutions of prescribing scalar curvature equations $\sigma_2(\kappa) = f(x)$ as well as the Hessian equations $\sigma_2(D^2u) = f(x)$. This new…
In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions $n\leq 4$, for all values of the Lagrangian phase, any $C^{1,1}$ solution is smooth and derive…
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to…
In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations with homogeneous boundary data in Minkowski space. By establishing the a priori C2 estimates, we obtain the existence…
We establish $C^{\sigma+\alpha}$ interior estimates for concave nonlocal fully nonlinear equations of order $\sigma\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^{\alpha}(\mathbb R^n)$ solves in $B_1$ a concave translation…
In this paper, We establish Pogorelov type $C^2$ estimates for the admissible solutions with $\sigma_k(D^2u)$ bounded from below of Sum Hessian equations. We also proved the lower bounded condition can be removed when $k = n$.
We show a second order a priori estimate for solutions to the complex $k$-Hessian equation on a compact K\"ahler manifold provided the $(k$-$1)$-st root of the right hand side is $\mathcal C^{1,1}$. This improves an estimate of Hou-Ma-Wu.…
The main result of this paper gives a plenary proof on the curvature estimates for $k$ curvature equations with general right hand sides with $n<2k$ based on a concavity inequality. We further give a explicit lower bound of the inequality.
In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve…
In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,\alpha}$ ($k\geq 1$, $0<\alpha<1$) regularity with smallness assumptions on the solution…
In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations, and establish the global C^1 estimates a nd reduce the global second derivative estimate to the estimate of double normal second derivatives on…
In this paper, we establish Pogorelov type $C^2$ estimates for admissible solutions to the Dirichlet problem of $(n-1)$-Hessian equation based on a concavity inequality, which is inspired by the Lu-Tsai's work on the global curvature…
In this paper, we discuss the more general Hessian inequality $\sigma_{k}^{\frac{1}{k}}(\lambda (D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and…
We establish sharp regularity estimates for solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric L\'evy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called…