论文标题
在有限域中的分数korn不平等中：案例的反例$ ps <1 $
On the fractional Korn inequality in bounded domains: Counterexamples to the case $ps<1$
论文作者
论文摘要
Korn在有限域中的分数环境中第一个不平等的有效性已经开放。我们通过证明Korn的第一个不等式在情况下$ ps> 1 $来解决这个问题，以$ w^{s，p} _0（ω）$ sobolev fields open and Bounded $ c^{1} $ - 常规域$ω\ subset $ω\ subset \ subset \ subset \ mathbb r^n $。同样，在任何开放界限的$ c^1 $域$ ps <1的情况下，$ $ $ω\ subset \ mathbb r^n $我们构造了不平等的反例，即，科恩的第一个不平等现象未能保留在有限的域中。 $ ps> 1 $的不平等的证据遵循经典案例中采用的标准紧凑型方法，加上耐寒性不平等，以及最近被Mengesha和Scott [\ textit {commun的Korn第二不平等现象。数学。 Sci。，} vol。 20，n0。 2，405--423，2022]。 $ ps <1 $在情况下构建的反例是远离边界内域内恒定仿射刚性运动的插值，靠近边界的零字段的插值。
The validity of Korn's first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn's first inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(Ω)$ Sobolev fields in open and bounded $C^{1}$-regular domains $Ω\subset \mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain $Ω\subset \mathbb R^n$ we construct counterexamples to the inequality, i.e., Korn's first inequality fails to hold in bounded domains. The proof of the inequality in the case $ps>1$ follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [\textit{Commun. Math. Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in the case $ps<1$ are interpolations of a constant affine rigid motion inside the domain away from the boundary, and of the zero field close to the boundary.
