关于均匀领域上的分数Korn不等式及截断半范数的新Korn不等式

BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension

In this manuscript two $BMO$ estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the $BMO$-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the $BMO$-seminorm of the symmetric part of its gradient, that is, a Korn inequality in $BMO$. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such a…

A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations

In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with 'projected difference' with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equat…

A truncated real interpolation method and characterizations of screened Sobolev spaces

In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.

重排与Maz'ya-Shaposhnikova极限：截断相互作用的Γ-收敛

Rearrangements and the Maz'ya-Shaposhnikova limit: $Γ$-convergence for truncated interactions

本文通过球面重排证明Maz'ya-Shaposhnikova极限的Γ-下极限不等式，并应用于截断能量，证明其Γ-收敛到(1-λ)Nω_N‖u‖_{L^2}^2，实现长程贡献识别与连续插值。

由分数阶梯度能量产生的拓扑奇点

Topological singularities arising from fractional-gradient energies

该研究证明平面正则区域上经标度的Ginzburg-Landau型泛函相对于雅可比平坦收敛Γ收敛于涡旋型能量，通过与含Riesz位势的标准泛函比较得紧性与Γ-下极限，用连接截断类涡旋函数的直接论证得Γ-上极限。

Fractional Sobolev spaces with power weights

We investigate the form of the closure of the smooth, compactly supported functions $C_{c}^{\infty}(Ω)$ in the weighted fractional Sobolev space $W^{s,p;\,w,v}(Ω)$ for bounded $Ω$. We focus on the weights $w,\,v$ being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of $Ω$. For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.