论文标题
$ l^p $ - 无兼容的korn不平等不平等的不兼容张量场的不平等现象
$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions
论文作者
论文摘要
对于$ 1 <p <\ infty $，我们证明了$ l^p $ - 不兼容的无跟踪的Korn不平等，用于不兼容的张量字段$ p $ in $ w^{1，\，\，\，p} _0（\ propatatOrnAme {curl}更准确地说，令$ω\ subset \ mathbb {r}^3 $为有界的Lipschitz域。然后存在一个常数$ c> 0 $，这样就可以，以至于\ [\ | {p} \ | _ {l^p（ω，\ mathbb {r}^{3 \ times3}）} \ leq c \ leq c \，\ left（ } \ | _ {l^p（ω，\ Mathbb {r}^{3 \ times3}）} + \ | {\ operatorName {dev} \ operatorname {curl} p} p} p}所有张量的字段$ p \ in w^{1，\，\，p} _0（\ operatotorName {curl};ω，\ mathbb {r}^{3 \ times3}）$，即，I.E. ω，\ mathbb {r}^{3 \ times3}）$带有损失的切向跟踪$ p \ times $ p \ times c $ \partialΩ \ operatotorname {tr}（p）\，\ mathbb {1} _3 $表示$ p $的偏离（无跟踪）部分。我们还显示规范等价\ [\ | {p} \ | _ {l^p（ω，\ mathbb {r}^{3 \ times3}）}+\ | {\ | {\ operatorname {curl} p} p} p} p} p} p} c \，\ left（\ | {\ operatOrname {dev} \ propatorAtorName {sym} p} \ | _ {l^p（ω，\ mathbb {r}^r}^{3 \ times3} {3 \ times3}} } \ | _ {l^p（ω，\ Mathbb {r}^{3 \ times3}）} \ right）\] \ right）\] \] \] \] \] \] \]这些估计值也适用于仅在边界的相对开放（非空）子集$γ\ subseteq \partialΩ$上的张量字段。
For $1<p<\infty$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $ W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$. More precisely, let $Ω\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$ with vanishing tangential trace $ P\times ν=0 $ on $ \partialΩ$ where $ν$ denotes the outward unit normal vector field to $\partialΩ$ and $\operatorname{dev} P := P -\frac13 \operatorname{tr}(P)\,\mathbb{1}_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; Ω,\mathbb{R}^{3\times3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $Γ\subseteq \partialΩ$ of the boundary.
