Nečas–Lions lemma revisited: An Lp‐version of the generalized Korn inequality for incompatible tensor fields
Abstract: For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in W01,0.1empfalse(Curl;normalΩ,ℝ3×3false). More precisely, let normalΩ⊂ℝ3 be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) > 0 such that ‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3) holds for all tensor fields P∈W01,0.1empfalse(Curl;normalΩ,ℝ3×3false), that is, for all P∈W1,0.1empfalse(Curl;normalΩ,ℝ3×3false) with vanishing tangential trace P×ν=0 on ∂Ω where ν denotes th… Show more
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Cited by 23 publications
References 67 publications
$$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions
For $$n\ge 3$$ 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.
$$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions
For $$n\ge 3$$ 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.
We establish a family of inequalities that allow one to estimate the L qnorm of a matrix-valued field by the L q -norm of an elliptic part and the L p -norm of the matrix-valued curl. This particularly extends previous work by Neff et al. and, as a main novelty, is applicable in the regime p = 1.
We characterise all linear maps $${\mathscr {A}}:\mathbb R^{n\times n}\rightarrow \mathbb R^{n\times n}$$ A : R n × n → R n × n such that, for $$1\le p<n$$ 1 ≤ p < n , $$\begin{aligned} \left\Vert P\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}\le c\,\Big (\left\Vert {\mathscr {A}}[P]\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}+\left\Vert {\text {Curl}}P\right\Vert _{{\text {L}}^{p}(\mathbb R^{n})} \Big ) \end{aligned}$$ P L p ∗ ( R n ) ≤ c ( A [ P ] L p ∗ ( R n ) + Curl P L p ( R n ) ) holds for all compactly supported $$P\in {\text {C}}_{c}^{\infty }(\mathbb R^{n};\mathbb R^{n\times n})$$ P ∈ C c ∞ ( R n ; R n × n ) , where $${\text {Curl}}P$$ Curl P displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of $${\mathscr {A}}$$ A , the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation.
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