On an Extension of the First Korn Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

Abstract

For a bounded domain $\varOmega$ in $\mathbb {R}^N$ with Lipschitz boundary $\varGamma =\partial \varOmega$ and a relatively open and non-empty subset $\varGamma _t$ of $\varGamma$, we prove the existence of a positive constant $c$ such that inequality $c\Vert T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})} \le \Vert {{\mathrm{sym}}}T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})} +\Vert {{\mathrm{Curl}}}T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N(N-1)/2})}$ holds for all tensor fields $T \in \overset{\circ }{\mathsf {H}}({{\mathrm{Curl}}};\varGamma _t,\varOmega ,\mathbb {R}^{N\times N})$, this is, for all $T:\varOmega \rightarrow \mathbb {R}^{N\times N}$ which are square-integrable and possess a row-wise square-integrable rotation tensor field ${{\mathrm{Curl}}}T:\varOmega \rightarrow \mathbb {R}^{N\times N(N-1)/2}$ and vanishing row-wise tangential trace on $\varGamma _t$. For compatible tensor fields $T=\nabla { v}$ with ${ v}\in \mathsf {H}^1(\varOmega ,\mathbb {R}^N)$ having vanishing tangential Neumann trace on $\varGamma _t$ the inequality reduces to a non-standard variant of the first Korn inequality since ${{\mathrm{Curl}}}T=0$, while for skew-symmetric tensor fields $T$ the Poincaré inequality is recovered. If $\varGamma _t=\emptyset$, our estimate still holds at least for simply connected $\varOmega$ and for all tensor fields $T \in \mathsf {H}({{\mathrm{Curl}}};\varOmega ,\mathbb {R}^{N\times N})$ which are $\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})$-perpendicular to ${{\mathrm{\mathfrak {so}}}}(N)$, i.e., to all skew-symmetric constant tensors.

Appendix: Construction of Hodge-Helmholtz Projections

We want to point out how to compute the projections in the Hodge-Helmholtz decompositions in Lemma 8.2. Recalling from Lemma 8.2 the orthogonal decompositions

$$\begin{aligned} \mathsf {L}^{2,q}(\varOmega )&= \mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega ) \oplus \overset{\circ }{\Delta }{}_0^q(\varGamma _n,\varOmega ) \\&= \overset{\circ }{\mathsf {D}}{}_0^q(\varGamma _t,\varOmega ) \oplus \delta \overset{\circ }{\Delta }{}^{q+1}(\varGamma _n,\varOmega ) \\&= \mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega ) \oplus \fancyscript{H}^q(\varOmega ) \oplus \delta \overset{\circ }{\Delta }{}^{q+1}(\varGamma _n,\varOmega ), \end{aligned}$$

we denote the corresponding $\mathsf {L}^{2,q}(\varOmega )$-orthogonal projections by $\pi _{\mathrm{{d}}}$, $\pi _{\delta }$ and $\pi _{\fancyscript{H}}$. Then, we have

$$ \pi _{\fancyscript{H}}={{\mathrm{id}}}-\pi _{\mathrm{{d}}}-\pi _{\delta } $$

and

$$\begin{aligned} \pi _{\mathrm{{d}}}\mathsf {L}^{2,q}(\varOmega )&= \mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega ) =\mathrm{{d}}\mathsf {X}^{q-1}(\varOmega ),&\mathsf {X}^{q-1}(\varOmega )&:= \overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega ) \cap \delta \overset{\circ }{\Delta }{}^q(\varGamma _n,\varOmega ), \\ \pi _{\delta }\mathsf {L}^{2,q}(\varOmega )&= \delta \overset{\circ }{\Delta }{}^{q+1}(\varGamma _n,\varOmega ) =\delta \mathsf {Y}^{q+1}(\varOmega ),&\mathsf {Y}^{q+1}(\varOmega )&:= \overset{\circ }{\Delta }{}^{q+1}(\varGamma _n,\varOmega ) \cap \mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^q(\varGamma _t,\varOmega ), \\ \pi _{\fancyscript{H}}\mathsf {L}^{2,q}(\varOmega )&= \fancyscript{H}^q(\varOmega ). \end{aligned}$$

By the Poincaré estimate, i.e., Lemma 8.1, we have

$$\begin{aligned} \forall E&\in \mathsf {X}^{q-1}(\varOmega ) \quad&\Vert E\Vert _{\mathsf {L}^{2,q-1}(\varOmega )}&\le c_{\mathtt {p},q-1}\Vert \mathrm{{d}}E\Vert _{\mathsf {L}^{2,q}(\varOmega )}, \end{aligned}$$

(8.9)

$$\begin{aligned} \forall H&\in \mathsf {Y}^{q+1}(\varOmega ) \quad&\Vert H\Vert _{\mathsf {L}^{2,q+1}(\varOmega )}&\le c_{\mathtt {p},q+1}\Vert \delta H\Vert _{\mathsf {L}^{2,q}(\varOmega )}. \end{aligned}$$

(8.10)

Hence, the bilinear forms

$$ \left( \tilde{E},E\right) \mapsto \left\langle \mathrm{{d}}\tilde{E},\mathrm{{d}}E \right\rangle _{\mathsf {L}^{2,q}(\varOmega )}, \quad \left( \tilde{H},H\right) \mapsto \left\langle \delta \tilde{H},\delta H \right\rangle _{\mathsf {L}^{2,q}(\varOmega )} $$

are continuous and coercive over $\mathsf {X}^{q-1}(\varOmega )$ and $\mathsf {Y}^{q+1}(\varOmega )$, respectively. Moreover, for any $F\in \mathsf {L}^{2,q}(\varOmega )$ the linear functionals

$$ E \mapsto \langle F,\mathrm{{d}}E \rangle _{\mathsf {L}^{2,q}(\varOmega )}, \quad H \mapsto \langle F,\delta H \rangle _{\mathsf {L}^{2,q}(\varOmega )} $$

are continuous over $\mathsf {X}^{q-1}(\varOmega )$, respectively $\mathsf {Y}^{q+1}(\varOmega )$. Thus, by the Lax-Milgram theorem we get unique solutions $E_{\mathrm{{d}}}\in \mathsf {X}^{q-1}(\varOmega )$ and $H_{\delta }\in \mathsf {Y}^{q+1}(\varOmega )$ of the two variational problems

$$\begin{aligned} \langle \mathrm{{d}}E_{\mathrm{{d}}},\mathrm{{d}}E \rangle _{\mathsf {L}^{2,q}(\varOmega )}&= \langle F,\mathrm{{d}}E \rangle _{\mathsf {L}^{2,q}(\varOmega )} \quad&\forall E\in \mathsf {X}^{q-1}(\varOmega ), \end{aligned}$$

(8.11)

$$\begin{aligned} \langle \delta H_{\delta },\delta H \rangle _{\mathsf {L}^{2,q}(\varOmega )}&= \langle F,\delta H \rangle _{\mathsf {L}^{2,q}(\varOmega )} \quad&\forall H\in \mathsf {Y}^{q+1}(\varOmega ) \end{aligned}$$

(8.12)

and the corresponding solution operators, mapping $F$ to $E_{\mathrm{{d}}}$ and $H_{\delta }$, respectively, are continuous. In fact, we have as usual

$$ \Vert \mathrm{{d}}E_{\mathrm{{d}}}\Vert _{\mathsf {L}^{2,q}(\varOmega )} \le \Vert F\Vert _{\mathsf {L}^{2,q}(\varOmega )}, \quad \Vert \delta H_{\delta }\Vert _{\mathsf {L}^{2,q}(\varOmega )} \le \Vert F\Vert _{\mathsf {L}^{2,q}(\varOmega )}, $$

respectively, and therefore together with (8.9) and (8.10)

$$\begin{aligned} \Vert E_{\mathrm{{d}}}\Vert _{\mathsf {X}^{q-1}(\varOmega )}&= \Vert E_{\mathrm{{d}}}\Vert _{\mathsf {D}^{q-1}(\varOmega )} \le \sqrt{1+c_{\mathtt {p},q-1}^2}\Vert F\Vert _{\mathsf {L}^{2,q}(\varOmega )},\\ \Vert H_{\delta }\Vert _{\mathsf {Y}^{q+1}(\varOmega )}&= \Vert H_{\delta }\Vert _{\varDelta ^{q+1}(\varOmega )} \le \sqrt{1+c_{\mathtt {p},q+1}^2}\Vert F\Vert _{\mathsf {L}^{2,q}(\varOmega )}. \end{aligned}$$

Since $\mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega )=\mathrm{{d}}\mathsf {X}^{q-1}(\varOmega )$ and $\delta \overset{\circ }{\varDelta }{}^{q+1}(\varGamma _n,\varOmega )=\delta \mathsf {Y}^{q+1}(\varOmega )$ we see that (8.11) and (8.12) hold also for $E\in \overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega )$ and $H\in \overset{\circ }{\varDelta }{}^{q+1}(\varGamma _n,\varOmega )$, respectively, and that

$$\begin{aligned} F-\mathrm{{d}}E_{\mathrm{{d}}}&\in \big (\mathrm{{d}}\mathsf {X}^{q-1}(\varOmega )\big )^{\bot } =\big (\mathrm{{d}}\overset{\circ }{\mathsf {D}}{}^{q-1}(\varGamma _t,\varOmega )\big )^{\bot }=\overset{\circ }{\varDelta }{}_0^q(\varGamma _n,\varOmega ),\\ F-\delta H_{\delta }&\in \big (\delta \mathsf {Y}^{q+1}(\varOmega )\big )^{\bot } =\big (\delta \overset{\circ }{\varDelta }{}^{q+1}(\varGamma _n,\varOmega )\big )^{\bot }=\overset{\circ }{\mathsf {D}}{}_0^q(\varGamma _t,\varOmega ). \end{aligned}$$

Hence, we have found our projections since

$$\begin{aligned} \pi _{\mathrm{{d}}}F&:= \mathrm{{d}}E_{\mathrm{{d}}}\in \mathrm{{d}}\mathsf {X}^{q-1}(\varOmega )\subset \overset{\circ }{\mathsf {D}}{}_0^q(\varGamma _t,\varOmega ),\\ \pi _{\delta }F&:= \delta H_{\delta }\in \delta \mathsf {Y}^{q+1}(\varOmega )\subset \overset{\circ }{\varDelta }{}_0^q(\varGamma _n,\varOmega ) \end{aligned}$$

and

$$ \pi _{\fancyscript{H}}F:=F-\mathrm{{d}}E_{\mathrm{{d}}}-\delta H_{\delta } \in \overset{\circ }{\mathsf {D}}{}_0^q(\varGamma _t,\varOmega )\cap \overset{\circ }{\Delta }{}_0^q(\varGamma _n,\varOmega )=\fancyscript{H}^q(\varOmega ). $$

Explicit formulas for the dimensions of $\fancyscript{H}^q(\varOmega )$ or explicit constructions of bases of $\fancyscript{H}^q(\varOmega )$ depending on the topology of the pair $(\varOmega ,\varGamma _t)$ can be found, e.g., in [21] for the case $\varGamma _t=\varGamma$ or $\varGamma _t=\emptyset$, or in [3] for the general case.