论文标题
在三个空间尺寸的不兼容张量场的KORN不平等现象，具有共形的脱位能
Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^3 $是一个开放式和边界的设置，带有Lipschitz边界和外部单位正常$ν$。对于$ 1 <p <\ infty $，我们建立了新的Banach Space $$中不兼容的张量字段的广义$ l^p $ -korn不平等的改进版本 w^{1，\，\，p，\，r} _0（\ operatorName {dev} \ propatorName {sym} \ propatorAtorName {curl};ω，\ mathbb r^{3 \ times3} \ operatorName {dev} \ protatorName {sym} \ propatatorName {curl} p \ in l^r（ω，\ mathbb r^{3 \ times3}），\\ \ \ \ \ operatorname {dev} $$在哪里 $ r \ in [1，\ infty），\ qquad \ frac1r \ le \ frac1p + \ frac13，\ qquad r> 1 \ quad \ text \ text {如果$ p = \ frac32 $。} $$ 具体来说，存在常数$ c = c（p，ω，r）> 0 $，使得不等式 \ [\ | p \ | _ {l^p} \ leq c \，\ left（\ | \ | \ operatoRatOrname {sym} p \ | _ {l^p} + \ | \ | \ | \ | \ | \ | \ operatorname {dev} \ protatorname {sym} \]保留所有张量字段$ p \ in W^{1，\，\，p，\，r} _0（\ operatatorName {dev} \ propatatorName {sym} \ operatoratorname {sym} \ operatotorname {curl}）$。在这里，$ \ operatorName {dev} x：= x- \ frac13 \ frac13 \ propatatorName {tr}（x）\，\ mathbb {1} $表示$ 3 \ times 3 $ 3 $ matrix $ x $的偏差（无跟踪（无跟踪）部分，边界条件和边界条件在适当的弱点上是理解的。
Let $Ω\subset \mathbb{R}^3$ be an open and bounded set with Lipschitz boundary and outward unit normal $ν$. For $1<p<\infty$ we establish an improved version of the generalized $L^p$-Korn inequality for incompatible tensor fields $P$ in the new Banach space $$ W^{1,\,p,\,r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl}; Ω,\mathbb R^{3\times3}) = \{ P \in L^p(Ω,\mathbb R^{3\times3}) \mid \operatorname{dev} \operatorname{sym} \operatorname{Curl} P \in L^r(Ω,\mathbb R^{3\times3}),\ \operatorname{dev} \operatorname{sym} (P \times ν) = 0 \text{ on $\partial Ω$}\} $$ where $$ r \in [1, \infty), \qquad \frac1r \le \frac1p + \frac13, \qquad r >1 \quad \text{if $p = \frac32$.}$$ Specifically, there exists a constant $c=c(p,Ω,r)>0$ such that the inequality \[ \|P \|_{L^p}\leq c\,\left(\|\operatorname{sym} P \|_{L^p} + \|\operatorname{dev}\operatorname{sym} \operatorname{Curl} P \|_{L^{r}}\right) \] holds for all tensor fields $P\in W^{1,\,p, \, r}_0(\operatorname{dev}\operatorname{sym}\operatorname{Curl})$. Here, $\operatorname{dev} X := X -\frac13 \operatorname{tr}(X)\,\mathbb{1}$ denotes the deviatoric (trace-free) part of a $3 \times 3$ matrix $X$ and the boundary condition is understood in a suitable weak sense.
