Existence of Smooth Solutions of the Navier-Stokes Equations

In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3]

On the exterior Dirichlet problem for Hessian quotient equations

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Amp\`{e}re equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.Comment: 35 page

Global W2,δW^{2,\delta} estimates for a type of singular fully nonlinear elliptic equations

We obtain global $W^{2,\delta}$ estimates for a type of singular fully nonlinear elliptic equations where the right hand side term belongs to $L^\infty$. The main idea of the proof is to slide paraboloids from below and above to touch the solution of the equation, and then to estimate the low bound of the measure of the set of contact points by the measure of the set of vertex points.Comment: 17 page

Existence and boundary asymptotic behavior of large solutions of Hessian equations

In this paper, we establish the existence of large solutions of Hessian equations and obtain a new boundary asymptotic behavior of solutions

A note on the Harnack inequality for elliptic equations in divergence form

In 1957, De Giorgi [3] proved the H\"{o}lder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3]

A Bernstein problem for special Lagrangian equations in exterior domains

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Amp\`{e}re equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.Comment: 20 page

An Optimal Geometric Condition on Domains for Boundary Differentiability of Solutions of Elliptic Equations

In this paper, a geometric condition on domains will be given which guarantees the boundary differentiability of solutions of elliptic equations, that is, the solutions are differentiable at any boundary point. We will show that this geometric condition is optimal

Regularity for fully nonlinear elliptic equations with oblique boundary conditions

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^{\alpha}$, $C^{1,\alpha}$ and $C^{2,\alpha}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations

Asymptotic behavior at infinity of solutions of Monge-Amp\`ere equations in half spaces

We prove that any convex viscosity solution of $\det D^2u=1$ outside a bounded domain of $\mathbb{R}^n_+$ tends to a quadratic polynomial at infinity with rate at least $\frac{x_n}{|x|^{n}}$ if $u$ is a quadratic polynomial on $\{x_n=0\}$ and satisfies $\mu|x|^2\leq u\leq \mu^{-1}|x|^2$ as $|x|\rightarrow \infty$ for some $0<\mu\leq \frac{1}{2}$.Comment: 26 page

Non-ergodic Complexity of Convex Proximal Inertial Gradient Descents

The proximal inertial gradient descent is efficient for the composite minimization and applicable for broad of machine learning problems. In this paper, we revisit the computational complexity of this algorithm and present other novel results, especially on the convergence rates of the objective function values. The non-ergodic O(1/k) rate is proved for proximal inertial gradient descent with constant stepzise when the objective function is coercive. When the objective function fails to promise coercivity, we prove the sublinear rate with diminishing inertial parameters. In the case that the objective function satisfies optimal strong convexity condition (which is much weaker than the strong convexity), the linear convergence is proved with much larger and general stepsize than previous literature. We also extend our results to the multi-block version and present the computational complexity. Both cyclic and stochastic index selection strategies are considered