Related papers: Subsolution theorem for the complex Hessian equati…

The main result of [HL] (Annals of Math. 102 (1975), 223--290) gives a characterization of the boundaries of complex subvarieties in C^n. An application of this result concerning pseudoconvexity, namely Theorem 10.4 in [HL], is amplified.

For solving large-scale non-convex problems, we propose inexact variants of trust region and adaptive cubic regularization methods, which, to increase efficiency, incorporate various approximations. In particular, in addition to approximate…

We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation $\sigma_2/\sigma_1=1$ in $\mathbb{R}^n$. The proof is based on the observation that after rewriting the quotient operator as the $\sigma_2$ operator,…

We show that for any $\epsilon\in ]0,1[$ there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball $B\subset\R^5$ which belongs to $C^{1,\epsilon} (B)\setminus C^{1,\epsilon+} (B)$.

We consider the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global…

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…

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a…

Let $X \subset \Bbb P^r$ be a smooth algebraic curve in projective space, over an algebraically closed field of characteristic zero. For each $m \in \Bbb N$, the $m$-flexes of $X$ are defined as the points where the osculating hypersurface…

In this paper we present two frameworks in which global maximization of a bounded hessian function over a strongly convex set can be reduced to convex optimization. The first presented framework is a continuation of one of our previous…

Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given…

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible hypersurfaces $H_1, H_2\subset\mathbb{C}^{n}$ whose complements $\mathbb{C}^{n}\setminus H_1$ and…

In this paper, we present a new form of the Hahn-Banach Theorem in terms of the sub-additive convex functions.

Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian…

Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and…

In this paper, we give a sufficient condition to guarantee the existence of a smooth solution of the Navier-Stokes Equation with the nice decreasing properties at infinity. In this way, we prove the existence of smooth physically reasonable…

This article is devoted to the study of lower semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians in a gradient variable. Such Hamiltonians appear in the optimal control theory. We present a necessary and…

We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the…

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 present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains…