2,211 research outputs found
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 estimates for a type of singular fully nonlinear elliptic equations
We obtain global estimates for a type of singular fully
nonlinear elliptic equations where the right hand side term belongs to
. 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 ,
and 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 outside a
bounded domain of tends to a quadratic polynomial at infinity
with rate at least if is a quadratic polynomial on
and satisfies as
for some .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
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