$$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

Abstract

AbstractFor $$n\ge 3$$ n ≥ 3 and $$1<p<\infty $$ 1 < p < ∞ , we prove an $$L^p$$ L p -version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields $$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$ P : Ω → R n × n having p-integrable generalized $${\text {Curl}}_{n}$$ Curl n and generalized vanishing tangential trace $$P\,\tau _l=0$$ P τ l = 0 on $$\partial \Omega $$ ∂ Ω , denoting by $$\{\tau _l\}_{l=1,\ldots , n-1}$$ { τ l } l = 1 , … , n - 1 a moving tangent frame on $$\partial \Omega $$ ∂ Ω . More precisely, there exists a constant $$c=c(n,p,\Omega )$$ c = c ( n , p , Ω ) such that $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$ ‖ P ‖ L p ( Ω , R n × n ) ≤ c ‖ dev n sym P ‖ L p ( Ω , R n × n ) + ‖ Curl n P ‖ L p Ω , R n × n ( n - 1 ) 2 , where the generalized $${\text {Curl}}_{n}$$ Curl n is given by $$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$ ( Curl n P ) ijk : = ∂ i P kj - ∂ j P ki and "Equation missing" denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.