Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性

In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^αΩ on a bounded open set Ω ■R(N ≥ 2) with C^(1,1) boundary ■Ω. We prove that when f ∈ L^p(Ω), and g ∈ C(Ω), the following problem (-△)^αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.     

形变张量的特征值与Boussinesq方程组的正则性估计

讨论了二维及三维满足周期边界条件的Boussinesq方程初边值问题的局部正则解在有限时间内爆破的可能性.在二维情况下,用形变张量的特征值给出温度梯度的L2估计,从中看出若流体微团变形的速率大,则解爆破的可能性就大.在三维情况下,用形变张量的特征值和温度的偏导给出涡量的L2估计,从中发现若流体微团在大部分时间内一般是平面拉伸,且温度的偏导较小时,解爆破的可能性就大;若一般是线性拉伸,温度的偏导又不任意增大时,解爆破的可能性就小.     

Regularity of Radial Solutions to the Complex Hessian Equations

In this paper we consider regularities of radial solutions to the degenerate complex Hessian equations. Our results generalize some results for Monge-Ampere equation in [Monn, Math. Ana. 275 （1986）, pp. 501-511] and [Delanoe, J. Diff. Eqn. 58 （1985）, pp. 318-344].     

p-Laplace方程正解的存在性和多重性

这篇文章讨论边值问题-(| u′|p-2u′)′=λf(t ,u) ,t∈(0,1) ,p >1,u(0) =u(1) =0,其中f(t ,u)≥-M( M是正常数) ,对(t ,u)∈0,1×0,∞) .我们利用度理论和锥上的不动点定理得到方程存在两个正解.     

一类非散度型椭圆方程解的局部W~(2,p)正则性

本文得到了一类具有小BMO系数的非散度型椭圆方程解的局部W2,p(p≥2) 正则性结果,并给出了这个结果的一个应用．     

非齐次p-调和映射方程组弱驻点解的正则性

讨论非齐次p－调和映射方程组弱驻点解的内部正则性，证明了弱驻点解满足拟单调不等式，得到了弱驻点解梯度的Holder连续性。     

A note on extinction for fast diffusive p‐Laplacian with sources

In this note we illuminate that the small condition on initial data u₀ in Theorem 4.1 of Yin and Jin (Math. Meth. Appl. Sci. 2007; 30 (10):1147–1167) can be removed for the case p?1<q<1. Precise decay estimates of solution are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and Regularity of Suitably Weak Solutions of n-dimensional Magnetohydrodynamics Equations

In this paper, it is shown that the weak solutions of magnetohydrodynamics equations in spaces L^q(R^+; L^p(R^n)) are stable and regular.     

A Remark on the Sobolev Regularity of Classical Solutions of Strongly Elliptic Equations

We prove that C^2,α(Ω ) solutions of problem (1.2) below are in H^m+2(Ω) for all m ∈ ?, if f and the coefficients are in H^m (Ω) n C^0,α (Ω ) Previously, this result was explicitly known only if m> n/2 (or if m = 0). A similar result holds for the quasi-linear equation (1.11) below.     

A Remark on the Sobolev Regularity of Classical Solutions to Uniformly Parabolic Equations

We prove that C^2+α,1+α/2 (Q?) solutions of problem (1.6) below are in a subspace H_c^m+2(Q) of H^{m+2,(m+2)/2(Q)} for all m ∈ ?, if f and the coefficients are in H_c^m(Q)∪C^α,α/2 (Q?). We apply this result to obtain global existence of Sobolev solutions to the quasilinear problem (1.26) below.     

Representation of the eigenvalues of the p ‐Laplacian by means of a surface integral

We prove a natural generalization to the p ‐Laplacian of the celebrated Rellich identity. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonexistence of Global Weak Solutions for Evolution Equations with Fractional Laplacian

Mathematical Notes - A nonlocal nonlinear parabolic equation with fractional Laplacian is considered. By means of the method of test functions, the nonexistence of nontrivial global weak solutions...     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

泛函极小与非线性椭圆方程解的局部正则性

利用Giaquinta和Giusti的嵌入不等式和Sobolev空间方法,证明了在更一般条件下的变分问题中的泛函极小与非线性椭圆方程弱解的局部正则性.     

Regularity of Solutions to Nonlinear Parabolic Equations with a Lower-Order Term

In this paper, we give some regularity results of solutions to nonlinear parabolic equations with a lower-order term and inregular data.     

具间断系数的N维拟线性椭圆方程弱解的正则性

讨论了具间断系数的N维拟线性椭圆方程. 利用估计和差分逼近方法,证明了弱解的一阶导数H\"{o}lder连续到方程系数间断的内边界.     

Two Regularity Criteria Via the Logarithm of the Weak Solutions to the Micropolar Fluid Equations

In this note, a logarithmic improved regularity criteria for the micropolar fluid equations are established in terms of the velocity field or the pressure in the homogeneous Besov space.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.