Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-05-24 , DOI: 10.1007/s00033-021-01550-6 Peter Lewintan , Patrizio Neff
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For \(n\ge 3\) and \(1<p<\infty \), we prove an \(L^p\)-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields \(P:\Omega \rightarrow \mathbb {R}^{n\times n}\) having p-integrable generalized \({\text {Curl}}_{n}\) and generalized vanishing tangential trace \(P\,\tau _l=0\) on \(\partial \Omega \), denoting by \(\{\tau _l\}_{l=1,\ldots , n-1}\) a moving tangent frame on \(\partial \Omega \). More precisely, there exists a constant \(c=c(n,p,\Omega )\) such that
$$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$where the generalized \({\text {Curl}}_{n}\) is given by \(({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}\) and
中文翻译：
$$ L ^ p $$ L p-任意维中不兼容张量字段的广义Korn不等式的无迹版本
对于\（nge 3 \）和\（1 <p <\ infty \），我们证明了针对不相容p可积张量的广义无迹Korn型不等式的（L ^ p \） -版本具有p可积的广义\（{\ text {Curl}} _ {n} \）和广义消失切向轨迹\（）的字段\（P：\ Omega \ rightarrow \ mathbb {R} ^ {n \ timesn } \）P （\ partial \ Omega \）上的P \，\ tau _l = 0 \），用\（\ {\ tau _l \} _ {l = 1，\ ldots，n-1} \）表示运动切线框架在\（\ partial \ Omega \）上。更确切地说，存在一个常数\（c = c（n，p，\ Omega）\）这样
$$ \ begin {aligned} \ Vert P \ Vert _ {L ^ p（\ Omega，\ mathbb {R} ^ {n \ times n}）} \ le c \，\ left（\ Vert {\ text {dev }} _ n {\ text {sym}} P \ Vert _ {L ^ p（\ Omega，\ mathbb {R} ^ {n \ times n}）} + \ Vert {\ text {Curl}} _ {n} P \ Vert _ {L ^ p \ left（\ Omega，\ mathbb {R} ^ {n \ times \ frac {n（n-1）} {2}} \ right）} \ right），\ end {aligned } $$广义\（{\ text {Curl}} _ {n} \）由\（（{{text {Curl}} _ {n} P）_ {ijk}给出：= \ partial _i P_ {kj} -\ partial _j P_ {ki} \）和
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