Bernstein <Emphasis Type="Italic">n</Emphasis>-widths of Besov embeddings on Lipschitz domains

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Bq0s＋t（Lp0（Ω））→Bq1s（Lp1（Ω））, t〉max{d（1/p0-1/p1）, 0}, 1 ≤ p0, p1, q0, q1 ≤∞,where Bq0s＋t（Lp0（Ω）） is a Besov space defined on the bounded Lipschitz domain Ω ？ Rd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.     

Lipschitz曲面上的Besov空间和Triebel-Lizorkin空间

利用Clifford分析工具,给出了Lipschitz曲面上Besov空间与Triebel-Lizorkin空间定义,并研究其特征刻划．     

Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations

In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ?³. Let ω _θ , J _θ and u _θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω _θ , J _θ ) ∈ L ^q (0, T; L ^p ) or (ω _θ , ▽(u _θ e _θ )) ∈ L ^q (0, T; L ^p ) with $\tfrac{3} {p} + \tfrac{2} {q} \leqslant 2$ , $\tfrac{3} {2} < p < \infty$ . In the endpoint case, one needs conditions $\left( {\omega _\theta ,J_\theta } \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ or $\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ .     

The Besov Space B^{{0,1}}_{1} on Domains

Let Ω be a bounded Lipschitz domain. Define B ^0,1 _{1, r} (Ω) = {f∈L ¹ (Ω): there is an F∈B ^0,1 ₁ (ℝ ⁿ ) such that F|_Ω = f} and B ^0,1 _{1 z} (Ω) = {f∈B ^0,1 ₁ (ℝ ⁿ ) : f = 0 on ℝ ⁿ }. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ ⁿ ₊. Received June 8, 2000, Accepted October 24, 2000     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the local solutions can be extended globally in time. Our solutions also belong to the Serrin class in the usual Lebesgue space. The method is based on the maximal Lorentz regularity theorem of the Stokes equations in the homogeneous Besov spaces. As an application, we may handle such singular data as the Dirac measure and the single layer potential supported on the sphere.     

Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.     

Partial regularity of finite Morse index solutions to the Lane-Emden equation

We prove regularity and partial regularity results for finite Morse index solutions u∈H¹(Ω)∩Lp(Ω) to the Lane-Emden equation −Δu=|u|^p−1u in Ω.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

具有低阶项的散度型椭圆方程的解在Morrey空间上的局部正则性

本文研究了具有低阶项的散度型椭圆方程-(a_(ij)u_(x_i))_(x_j)+b_iu_(x_i)-(d_ju)_(x_j)+cu= (f_j)_(x_j),a．e．x∈Ω的解在Morrey空间上的局部正则性,其中a_(ij)∈VMO∩L~∞(Ω),低阶项系数属于适当的Morrey空间．     

A discrete norm on a Lipschitz surface and the Sobolev straightening of a boundary

Let a piece of the boundary of a Lipschitz domain be parameterized conventionally and let the traces of functions in the Sobolev space W _p ² be written out through this parameter. In this space, we propose a discrete (diadic) norm generalizing A. Kamont’s norm in the plane case. We study the conditions when the space of traces coincides with the corresponding space for the plane boundary.     

Conditional Regularity of Weak Solutions to the 3D Magnetic Bénard Fluid System

This paper concerns about the regularity conditions of weak solutions tothe magnetic Bénard fluid system in $mathbb{R}^3$. We show that a weak solution $(u,b,θ)(·,t)$ ofthe 3D magnetic Bénard fluid system defined in $[0,T),$ which satisfies some regularityrequirement as $(u,b,θ),$ is regular in $mathbb{R}^3×(0,T)$ and can be extended as a $C^∞$ solutionbeyond $T$.     

Spectral properties of the layer potentials associated with elasticity equations and transmission problems on Lipschitz domains

We study the invertibility of βI+K and βI+K^′ in L²(∂Ω) for where K,K^′ are double layer potentials related to elasticity equations and Ω is bounded Lipschitz domain in Rn. Consequently, the spectrum on real line lies in . Applications to transmission problems are also presented.     

Lipschitz Invariance of Critical Exponents on Besov Spaces

In this paper we prove that the critical exponents of Besov spaces on a compact set possessing an Ahlfors regular measure is an invariant under Lipschitz transforms. Under mild conditions, the critical exponent of Besov spaces of certain self-similar sets coincides with the walk dimension, which plays an important role in the analysis on fractals. As an application, we show examples having different critical exponents are not Lipschitz equivalent.