Blow-up rates for the general curve shortening flow

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in R².     

Explicit error estimate for the nonconforming wilson's element

In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson's element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.     

One-dimensional conformal metric flow

In this paper we continue our studies of the one-dimensional conformal metric flows, which were introduced in [Y. Ni, M. Zhu, Extremal one-dimensional conformal metrics, Commun. Contemp. Math., in press]. We prove the global existence and convergence of the flows, as well as the exponential convergence of the metrics under these flows.     

The Harnack estimate for a nonlinear parabolic equation under the Ricci flow

Let (M,g(t)), 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| ≤ C/t for some constant C > 0 and g(t) evolving by the Ricci flow

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e ^h (x) dV (x) the weighted measure and ??_??,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ \Delta _{\mu ,p} u = - \lambda _{\mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..     

Lower Estimates of the Norms of Extension Operators for Sobolev Spaces on the Halfline

It is proved that for an arbitrary extension operator its norm cannot be less than ε₀2^mm^—1/2p, ε₀ > 0. Previously only the upper estimates (≤ 8^m) for these norms were known.     

Gradient estimate for solutions to Poisson equations in metric measure spaces

Let (X,d) be a complete, pathwise connected metric measure space with a locally Ahlfors Q-regular measure μ, where Q>1. Suppose that (X,d,μ) supports a (local) (1,2)-Poincaré inequality and a suitable curvature lower bound. For the Poisson equation Δu=f on (X,d,μ), Moser-Trudinger and Sobolev inequalities are established for the gradient of u. The local Hölder continuity with optimal exponent of solutions is obtained.     

Riemann流形上改进的p-Laplace方程的梯度估计

本文研究Riemann流形上的改进的p-Laplace方程,运用截断函数的估计、Hessian比较定理和Laplace比较定理,得到该方程正解的梯度估计.并应用该结论,得到在Riemann流形上关于改进的p-Laplace方程正解的Harnack不等式和Liouville型定理.     

完备Khler流形上Khler-Ricci流的局部Harnack估计

本文首先给出非正规化Khler-Ricci流下曲率的发展方程,然后得到了关于曲率的Harnack量在满足曲率局部条件下所产生的一个特殊项CNS.通过对CNS的估计,得到了完备Khler流形上关于Khler-Ricci流的局部Harnack不等式.最后,作为主要定理的应用,我们将结果推广到数量曲率的情形.     

On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations

We give a complete proof of Morrey’s estimate for the W ^1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L ₁-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.     

Cea's error estimate for strongly monotone variational inequalities

Céa's approximation lemma is extended to variational inequalities which are defined by strongly monotone operators in closed convex subsets of linear normed spaces. This abstract error estimate is applied to the finite element discretization of a nonlinear elliptic two-sided obstacle problem providing an asymptotic error estimate for a smooth enough solution.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

Weighted L~p-estimates for Stokes flow in R_+~n with applications to the non-stationary Navier-Stokes flow

We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for the Stokes semigroup generated by the Stokes operator in the half-space R+n (n 2). As an application of the weighted estimates of the Stokes semigroup, a class of local and global strong solutions in weighted Lp (R+n) are constructed, following the approach given by Kato.     

ON CRITICAL CASES OF SOBOLEV’S INEQUALITIES FOR HEISENBERG GROUPS

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.     

A rearrangement estimate for the generalized multilinear fractional integrals

We study the boundedness of generalized multilinear fractional integrals. An O’Neil type inequality for a k-linear integral operator is proved. Using an O’Neil type inequality for a k-linear integral operator, we obtain a pointwise rearrangement estimate of generalized multilinear fractional integrals. By way of application we prove a Sobolev type theorem for these integrals. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 3, pp. 577–585, May–June, 2007.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.