Related papers: Interior Hessian estimates for sum Hessian quotien…
In this paper, we consider the entire solutions to the parabolic $2$-Hessian equations of the form $-u_t\sigma_2(D^2 u)=1$ in $\mathbb{R}^n\times (-\infty,0]$. We prove some rigidity theorems for the parabolic $2$-Hessian equations in…
In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed…
We prove interior $H^{2s-\varepsilon}$ regularity for weak solutions of linear elliptic integro-differential equations close to the fractional $s$-Laplacian. The result is obtained via intermediate estimates in Nikol'skii spaces, which are…
We derive Hessian estimates for convex solutions to quadratic Hessian equation by a compactness argument.
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…
Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…
In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{\sigma_k(\Lambda(D^2u))}{\sigma_l(\Lambda(D^2u))}$, which can be regarded as a generalization of the classical Hessian quotient…
In this paper, we consider a Class of Hessian quotient equations in Euclidean space. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the…
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 prove interior Poincar{\'e} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike…
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed K\"ahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which…
New, doubling proofs are given for the interior Hessian estimates of the special Lagrangian equation. These estimates were originally shown by Chen-Warren-Yuan in CPAM 2009 and Wang-Yuan in AJM 2014. This yields a higher codimension…
We modify Pogorelov's classic construction to demonstrate the absence of a priori $C^2$ estimates for the equations $\det(D^2 u \pm Du \otimes Du) = f(x)$ in dimension $n \ge 3$. We construct a sequence of solutions $z_\varepsilon$ with…
In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type…
We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.
We obtain weighted mixed norm Sobolev estimates in the whole space for nonstationary Stokes equations in divergence and nondivergence form with variable viscosity coefficients that are merely measurable in time variable and have small mean…
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…
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…
The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss…
We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindel\"of hypothesis and a slightly…