Related papers: $C^{2,\alpha}$ estimates for nonlinear elliptic eq…
We obtain pointwise $C^{2,\alpha}$ estimates at boundary points for solutions to the Monge-Ampere equation under appropriate local conditions on the right hand side and boundary data.
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…
We extend the Caffarelli-\'Swiech-Winter $C^{1,\alpha}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded…
In this article, we prove the local $C^{0,\alpha}$ regularity and provide $C^{0,\alpha}$ estimates for viscosity solutions of fully nonlinear, possibly degenerate, elliptic equations associated to linear or nonlinear Neumann type boundary…
We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local $C^{1,\alpha}$ estimates on each side of the smooth obstacle, for some small $\alpha > 0$. Our results extend those of Milakis-Silvestre…
We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Amp\`ere equations $$\det D^2u =f(x) \text{ in } \Omega , \quad D_{\beta}u = \phi(x) \text{ on } \partial \Omega.$$ In…
In this paper, we establish pointwise boundary ${{C}^{1,\alpha}}$ estimates for viscosity solutions of some degenerate fully nonlinear elliptic equations on ${C^{1,\alpha}}$ domains. Instead of straightening out the boundary, we utilize the…
We establish sharp $C^{2s}$ interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right hand side. More precisely, we show that if $I$ is a fully nonlinear nonlocal concave or convex elliptic…
We classify global solutions of the Monge-Amp\`ere equation $\det D^2 u=1 $ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^{2,\alpha}$ estimates for the non-degenerate Monge-Amp\`ere…
In this article we establish sharp $C^{1,\alpha}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying…
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 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.
In this paper, we establish $C^{1, \alpha}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the…
We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent…
This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an…
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…
In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations…
In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates…
We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$…
We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior…