Gradient Estimates in Fractional Dirichlet Problems

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

     

具有双参数的弱非线性方程的奇摄动解

讨论了一四阶具有双参数的弱非线性方程在有限区间上的奇摄动边值问题．在一定的假设下,首先,利用幂级数形式展开方法,构造了原问题的外部解A·D2其次,利用伸长变量,在左端点附近构造问题解的第一边界层校正项．然后,利用更强的伸长变量,仍然在左端点附近构造问题解的第二边界层校正项．第二边界层的厚度比第一边界层的厚度更小,形成在左端点附近的边界层的套层．最后利用微分不等式理论,证明了边值问题解的存在性、和在整个区间内一致有效性和渐近性态,得到了满意的结果．     

Boundary behaviors for Liouville's equation in planar singular domains

We study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary. The conformal structure plays an essential role.     

Local regularity of the Navier-Stokes equations near the curved boundary

We present some regularity conditions for suitable weak solutions of the Navier-Stokes equations near the curved boundary of a sufficiently smooth domain. Our extend the work that was results established near a flat boundary by Gustafson, Kang and Tsai (2006) [6].     

<Emphasis Type="Italic">L</Emphasis><Subscript>3,∞</Subscript>-solutions to the 3D-Navier-Stokes system in a domain with a curved boundary

We show that L_3,∞-solutions to the three-dimensional Navier-Stokes equations near a curved smooth part of the boundary are Hölder continuous. The corresponding result near a planar part of the boundary was obtained earlier by G. Seregin. Bigliography: 22 titles.     

Probabilistic Proof of Boundary Harnack Principle for α-Harmonic Functions

The boundary Harnack principle for fractional Laplacian is now known to hold in Lipschitz domains [5]. It states that if two nonnegative functions, harmonic with respect to a symmetric stable Lévy process vanish continuously outside a Lipschitz domain, near a part of its boundary, then the ratio of the functions is bounded inside the domain, near this part of the boundary. We give a probabilistic proof of the assertion using elementary properties of the stable process.     

Cooperation of deterministic and stochastic mechanisms resulting in the intermittent behavior

Intermittent behavior near the boundary of chaotic phase synchronization in the presence of noise (when deterministic and stochastic mechanisms resulting in intermittency take place simultaneously) is studied. The noise of small intensity is shown to do not affect on the characteristics of intermittency whereas the noise of large amplitude induces new effects near the boundary of the synchronous regime. In the first case the eyelet intermittency takes place near the boundary of the synchronous regime, in the second one the ring intermittency or coexistence of both types of intermittency is realized. Main results are illustrated using the example of two unidirectionally coupled Rössler systems. Similar effects are shown to be observed in coupled spatially distributed Pierce beam–plasma systems.     

Vector fields near a generic submanifold

We give a classification of generic vector fields near a generic submanifold. The normal forms are linear vector fields near the local model of the submanifold. Similar results are obtained for vector fields near a hypersurface with boundary and near a piecewise-smooth hypersurface.     

边界伸缩法

本文提出边界伸缩原理,并在此基础上提出边界伸缩法.不但较好地解决了边界元方法中求解边界附近区域包括边界上解的问题,而且可以方便地利用迭代过程改善求解精度.计算实例表明.本文提出的方法是十分有效的.     

An anomaly formula for Ray–Singer metrics on manifolds with boundary

Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a non-product metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary. Received: January 2004; Revision: February 2005; Accepted: September 2005     

On the boundary regularity of weak solutions to the MHD system

The partial regularity of boundary suitable weak solutions to the MHD system near the plane part of the boundary is proved. Bibliography: 16 titles.     

Improved Exponential Convergence Rates by Oversampling Near the Boundary

Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.     

Boundary layers for stress diffusive perturbation in viscoelastic fluids

In this paper, we study a stress diffusive perturbation of the system describing a viscoelastic flow. We analyse the boundary layer which arises near the boundary and we observe in particular that there is no boundary layer on the velocity at the first order.     

Optimal boundary regularity for nonlinear singular elliptic equations

In this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary. By a careful construction of super- and sub-solutions, we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument.     

Porosity of the Free Boundary for p-Parabolic Type Equations in Non-Divergence Form

Potential Analysis - In this article we establish the exact growth of the solution to the singular quasilinear p-parabolic free boundary problem in non-divergence form near the free boundary from...     

GLOBAL SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE HEAT-CONDUCTING FLOW WITH SYMMETRY AND FREE BOUNDARY

ABSTRACT

Global solutions of the multidimensional Navier-Stokes equations for compressible heat-conducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heat-conducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

理想弹塑性I型平面应力裂纹线场的精确解

本文纠正了过去在裂纹弹塑性场匹配上存在的问题,采用线场分析方法,通过求得塑性区应力场的合理解答,使之与弹性精确场在裂纹线附近的弹塑性边界上匹配。本文就远场受单向拉伸及双向拉伸的理想弹塑性平面应力裂纹无限板,在完全放弃了小范围屈服条件的情况下求得了塑性区应力场、塑性区长度以及弹塑性边界的单位法向量在裂纹线附近足够精确的表达式。结果表明,无论单向拉伸和双向拉伸,塑性区应力分量σ_y,σ_xy,塑性区长度以及弹塑性边界的单位法向量在裂纹线附近的表达式完全相同,但塑性区沿X方向的正应力σ_x存在差别。