On the Maxwell and Friedrichs/Poincaré constants in ND

Abstract

We prove that for bounded and convex domains in arbitrary dimensions, the Maxwell constants are bounded from below and above by Friedrichs’ and Poincaré’s constants, respectively. Especially, the second positive Maxwell eigenvalues in ND are bounded from below by the square root of the second Neumann-Laplace eigenvalue.

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Acknowledgements

We cordially thank the anonymous referee for a very careful reading and valuable suggestions for improving the paper.

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Appendices

Appendix A: Proof of Lemma 3.2

By the \(*\)-operator it is sufficient to discuss, e.g., \(\omega \in \mathsf {D}^{q}(\Omega )\cap \mathring{\Delta }^{q}(\Omega )\). For a proof we follow the nice book of Grisvard, see [7, Theorem 3.2.1.2, Theorem 3.2.1.3]. This proof has been carried out in [5, Corollary 3.6, Theorem 3.9] and [1, Theorem 2.17] for the Maxwell case and \(N=3\). Our proof will avoid the misleading notion of traces and solutions of second order elliptic systems. Let us note that in [1, p. 834] the proof for \(X_{N}(\Omega )\) is wrong. One cannot work in the space \(V_{T}(\Omega _{k})\) due to the solenoidal condition. Working in the space \(X_{T}(\Omega _{k})\) is needed, but this destroys their argument for the second order elliptic system for \(\zeta \). Our approach corrects these inconsistencies.

Let us pick a sequence of increasing, convex, and \(\overset{}{\mathsf {C}}{}^{\infty }_{}\)-smooth subdomains \((\Omega _{n})\subset \Omega \) converging to \(\Omega \), i.e.,

$$\begin{aligned}\Omega _{n}\subset \overline{\Omega }_{n}\subset \Omega _{n+1}\subset \dots \subset \Omega ,\qquad {{\,\mathrm{dist}\,}}(\Omega ,\Omega _{n})={{\,\mathrm{dist}\,}}(\partial \Omega ,\partial \Omega _{n})\rightarrow 0,\end{aligned}$$

see, e.g., [7, Lemma 3.2.1.1]. Of course, \(\overset{}{\mathsf {C}}{}^{2}_{}\)-smooth is also sufficient. For \(\Omega _{n}\) we find \(\zeta _{n}\in \mathsf {D}^{q-1}(\Omega _{n})\) such that for all \(\varphi \in \mathsf {D}^{q-1}(\Omega _{n})\)

$$\begin{aligned} \langle \zeta _{n},\varphi \rangle _{\mathsf {D}^{q-1}(\Omega _{n})} =\langle \delta \omega ,\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})} +\langle \omega ,{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega _{n})}, \end{aligned}$$
(A.1)

which is a trivially well defined problem. Note \(\langle \zeta _{n},\varphi \rangle _{\mathsf {D}^{q-1}(\Omega _{n})} =\langle \zeta _{n},\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})} +\langle {{\,\mathrm{d}\,}}\zeta _{n},{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega _{n})}\). Hence

$$\begin{aligned} \langle \omega -{{\,\mathrm{d}\,}}\zeta _{n},{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega _{n})} =\langle \zeta _{n}-\delta \omega ,\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})} \end{aligned}$$

for all \(\varphi \in \mathsf {D}^{q-1}(\Omega _{n})\), showing by (2.11) that \(\omega _{n}:=\omega -{{\,\mathrm{d}\,}}\zeta _{n}\in \mathring{\Delta }^{q}(\Omega _{n})\) and \(\delta \omega _{n}=\delta \omega -\zeta _{n}\). Moreover, \(\omega _{n}\in \mathsf {D}^{q}(\Omega _{n})\) with \({{\,\mathrm{d}\,}}\omega _{n}={{\,\mathrm{d}\,}}\omega \). By (1.20) we have \(\omega _{n}\in \mathsf {H}^{1,q+1}(\Omega _{n})\) with

$$\begin{aligned} |\nabla \vec {\omega }_{n}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})}^2&\le |{{\,\mathrm{d}\,}}\omega _{n}|_{\mathsf {L}^{2,q+1}(\Omega _{n})}^2 +|\delta \omega _{n}|_{\mathsf {L}^{2,q-1}(\Omega _{n})}^2 =|{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega _{n})}^2 +|\delta \omega -\zeta _{n}|_{\mathsf {L}^{2,q-1}(\Omega _{n})}^2. \end{aligned}$$
(A.2)

By setting \(\varphi =\zeta _{n}\) in (A.1) we see

$$\begin{aligned} |\zeta _{n}|_{\mathsf {D}^{q-1}(\Omega _{n})}^2&=\langle \delta \omega ,\zeta _{n}\rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})} +\langle \omega ,{{\,\mathrm{d}\,}}\zeta _{n}\rangle _{\mathsf {L}^{2,q}(\Omega _{n})}\nonumber \\&\le |\delta \omega |_{\mathsf {L}^{2,q-1}(\Omega _{n})}|\zeta _{n}|_{\mathsf {L}^{2,q-1}(\Omega _{n})} +|\omega |_{\mathsf {L}^{2,q}(\Omega _{n})}|{{\,\mathrm{d}\,}}\zeta _{n}|_{\mathsf {L}^{2,q}(\Omega _{n})}\nonumber \\&\le |\omega |_{\Delta ^{q}(\Omega _{n})}|\zeta _{n}|_{\mathsf {D}^{q-1}(\Omega _{n})} \end{aligned}$$
(A.3)

and thus

$$\begin{aligned} |\zeta _{n}|_{\mathsf {D}^{q-1}(\Omega _{n})}&\le |\omega |_{\Delta ^{q}(\Omega _{n})} \le |\omega |_{\Delta ^{q}(\Omega )}. \end{aligned}$$
(A.4)

Combining (A.2) and the equation part of (A.3) we observe

$$\begin{aligned} |\vec {\omega }_{n}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega _{n})}^2&=|\omega _{n}|_{\mathsf {L}^{2,q}(\Omega _{n})}^2 +|\nabla \vec {\omega }_{n}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})}^2 \le |\omega _{n}|_{\mathsf {L}^{2,q}(\Omega _{n})}^2 +|{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega _{n})}^2 \\&\qquad +|\delta \omega -\zeta _{n}|_{\mathsf {L}^{2,q-1}(\Omega _{n})}^2\\&=|\omega |_{\mathsf {L}^{2,q}(\Omega _{n})}^2 +|{{\,\mathrm{d}\,}}\zeta _{n}|_{\mathsf {L}^{2,q}(\Omega _{n})}^2 +|{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega _{n})}^2 +|\delta \omega |_{\mathsf {L}^{2,q-1}(\Omega _{n})}^2 +|\zeta _{n}|_{\mathsf {L}^{2,q-1}(\Omega _{n})}^2\\&\qquad -2\langle \omega ,{{\,\mathrm{d}\,}}\zeta _{n}\rangle _{\mathsf {L}^{2,q}(\Omega _{n})} -2\langle \delta \omega ,\zeta _{n}\rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})}\\&=|\omega |_{\mathsf {D}^{q}(\Omega _{n})\cap \Delta ^{q}(\Omega _{n})}^2 +|\zeta _{n}|_{\mathsf {D}^{q}(\Omega _{n})}^2 -2|\zeta _{n}|_{\mathsf {D}^{q}(\Omega _{n})}^2 \le |\omega |_{\mathsf {D}^{q}(\Omega _{n})\cap \Delta ^{q}(\Omega _{n})}^2 \end{aligned}$$

and therefore

$$\begin{aligned} \begin{aligned} |\vec {\omega }_{n}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega _{n})} \le |\omega |_{\mathsf {D}^{q}(\Omega _{n})\cap \Delta ^{q}(\Omega _{n})} \le |\omega |_{\mathsf {D}^{q}(\Omega )\cap \Delta ^{q}(\Omega )}. \end{aligned} \end{aligned}$$
(A.5)

Let us denote the extension by zero to \(\Omega \) by \(\tilde{\cdot }\). Then by (A.4) and (A.5) the sequences \((\tilde{\zeta }_{n})\), \((\widetilde{{{\,\mathrm{d}\,}}\zeta }_{n})\), and \((\tilde{\vec {\omega }}_{n})\), \((\widetilde{\nabla \vec {\omega }}_{n})\) are bounded in \(\mathsf {L}^{2,q-1}(\Omega )\), \(\mathsf {L}^{2,q}(\Omega )\), resp. \(\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )\) and we can extract weakly converging subsequences, again denoted by the index n, such that

Let \(\psi \in \mathring{\mathsf {C}}^{\infty }(\Omega )\) and n be large enough such that \({{\,\mathrm{supp}\,}}\psi \subset \Omega _{n}\). Then \(\psi \in \mathring{\mathsf {C}}^{\infty }(\Omega _{n})\) and we calculate for \(i=1,\dots ,N\) and the \(\ell \)-th component \(\vec {\hat{\omega }}_{\ell }\) of \(\vec {\hat{\omega }}\)

$$\begin{aligned}&\langle \vec {\hat{\omega }}_{\ell },\partial _{i}\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} \leftarrow \langle \tilde{\vec {\omega }}_{n,\ell },\partial _{i}\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} =\langle \vec {\omega }_{n,\ell },\partial _{i}\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})}\\&\quad =-\langle \partial _{i}\vec {\omega }_{n,\ell },\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})} =-\langle \widetilde{\partial _{i}\vec {\omega }}_{n,\ell },\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}\rightarrow -\langle \hat{\Theta }_{i,\ell },\psi \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}, \end{aligned}$$

yielding \(\vec {\hat{\omega }}\in \overset{}{\mathsf {H}}{}^{1}_{}(\Omega )\) and \(\nabla \vec {\hat{\omega }}=\hat{\Theta }\). Analogously we obtain for \(\phi \in \mathring{\mathsf {C}}^{\infty ,q}(\Omega )\) with \(\phi \in \mathring{\mathsf {C}}^{\infty ,q}(\Omega _{n})\) for n large enough

$$\begin{aligned}&\langle \zeta ,\delta \phi \rangle _{\mathsf {L}^{2,q-1}(\Omega )} \leftarrow \langle \tilde{\zeta }_{n},\delta \phi \rangle _{\mathsf {L}^{2,q-1}(\Omega )} =\langle \zeta _{n},\delta \phi \rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})}\\&\quad =-\langle {{\,\mathrm{d}\,}}\zeta _{n},\phi \rangle _{\mathsf {L}^{2,q}(\Omega _{n})} =-\langle \widetilde{{{\,\mathrm{d}\,}}\zeta }_{n},\phi \rangle _{\mathsf {L}^{2,q}(\Omega )} \rightarrow -\langle \xi ,\phi \rangle _{\mathsf {L}^{2,q}(\Omega )}, \end{aligned}$$

showing \(\zeta \in \mathsf {D}^{q-1}(\Omega )\) and \({{\,\mathrm{d}\,}}\zeta =\xi \). Moreover, for \(\varphi \in \mathsf {D}^{q-1}(\Omega )\subset \mathsf {D}^{q-1}(\Omega _{n})\) we have by (A.1)

$$\begin{aligned}&\langle \zeta ,\varphi \rangle _{\mathsf {D}^{q-1}(\Omega )}\\&\quad =\langle \zeta ,\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega )} +\langle {{\,\mathrm{d}\,}}\zeta ,{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega )} \leftarrow \langle \tilde{\zeta }_{n},\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega )} +\langle \widetilde{{{\,\mathrm{d}\,}}\zeta }_{n},{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega )}\\&\quad =\langle \zeta _{n},\varphi \rangle _{\mathsf {D}^{q-1}(\Omega _{n})}=\langle \delta \omega ,\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega _{n})} +\langle \omega ,{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega _{n})}\\&\quad \rightarrow \langle \delta \omega ,\varphi \rangle _{\mathsf {L}^{2,q-1}(\Omega )} +\langle \omega ,{{\,\mathrm{d}\,}}\varphi \rangle _{\mathsf {L}^{2,q}(\Omega )} =0, \end{aligned}$$

as \(\omega \in \mathring{\Delta }^{q}(\Omega )\), where the last convergence follows by Lebesgue’s dominated convergence theorem. For \(\varphi =\zeta \) we get \(|\zeta |_{\mathsf {D}^{q-1}(\Omega )}=0\), i.e., \(\zeta =0\). Furthermore, we observe by (A.5)

$$\begin{aligned} |\vec {\hat{\omega }}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega )}^2&=\langle \vec {\hat{\omega }},\vec {\hat{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} +\langle \nabla \vec {\hat{\omega }},\nabla \vec {\hat{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} \leftarrow \langle \vec {\hat{\omega }},\tilde{\vec {\omega }}_{n}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} +\langle \nabla \vec {\hat{\omega }},\widetilde{\nabla \vec {\omega }}_{n}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}\\&=\langle \vec {\hat{\omega }},\vec {\omega }_{n}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})} +\langle \nabla \vec {\hat{\omega }},\nabla \vec {\omega }_{n}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega _{n})} \le |\vec {\hat{\omega }}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega _{n})} |\vec {\omega }_{n}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega _{n})}\\&\le |\vec {\hat{\omega }}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega )} |\omega |_{\mathsf {D}^{q}(\Omega )\cap \Delta ^{q}(\Omega )}, \end{aligned}$$

showing

$$\begin{aligned} |\vec {\hat{\omega }}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega )} \le |\omega |_{\mathsf {D}^{q}(\Omega )\cap \Delta ^{q}(\Omega )}. \end{aligned}$$
(A.6)

Finally, we have \(\omega =\omega _{n}+{{\,\mathrm{d}\,}}\zeta _{n}\) in \(\Omega _{n}\), i.e., in \(\Omega \)

On the other hand, by Lebesgue’s dominated convergence theorem we see \(\chi _{\Omega _{n}}\omega \rightarrow \omega \) in \(\mathsf {L}^{2,q}(\Omega )\). Thus \(\omega =\hat{\omega }\in \mathsf {H}^{1,q}(\Omega )\) and by (A.6)

$$\begin{aligned} |\omega |_{\mathsf {H}^{1,q}(\Omega )}&=|\vec {\hat{\omega }}|_{\overset{}{\mathsf {H}}{}^{1}_{}(\Omega )} \le |\omega |_{\mathsf {D}^{q}(\Omega )\cap \Delta ^{q}(\Omega )}, \end{aligned}$$

especially,

$$\begin{aligned} |\nabla \vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2&\le |{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega )}^2 +|\delta \omega |_{\mathsf {L}^{2,q-1}(\Omega )}^2. \end{aligned}$$

Appendix B: Calculations for Remark 3.12

For a multi index I of length \(|I|=q\) (not necessarily ordered) it holds

$$\begin{aligned} \Phi ^{*}{{\,\mathrm{d}\,}}x^{I}&=\Phi ^{*}({{\,\mathrm{d}\,}}x^{i_{1}}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{i_{q}}) =(\Phi ^{*}{{\,\mathrm{d}\,}}x^{i_{1}})\wedge \dots \wedge (\Phi ^{*}{{\,\mathrm{d}\,}}x^{i_{q}}) \\&=({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}}) = {{\,\mathrm{d}\,}}\Phi ^{I} \\&=\sum _{j_{1},\dots ,j_{q}}\partial _{j_{1}}\Phi _{i_{1}}\dots \partial _{j_{q}}\Phi _{i_{q}} {{\,\mathrm{d}\,}}x^{j_{1}}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{j_{q}} =\sum _{|J|=q}\partial _{J}\Phi _{I}{{\,\mathrm{d}\,}}x^{J} \end{aligned}$$

and especially

$$\begin{aligned}\Phi ^{*}({{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}) =\det (\nabla \Phi ){{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}.\end{aligned}$$

For multi indices IJ of length q we have

$$\begin{aligned} (\Phi ^{*}{{\,\mathrm{d}\,}}x^{I})\wedge *(\Phi ^{*}{{\,\mathrm{d}\,}}x^{J})&=\sum _{|K|=|L|=q}\partial _{K}\Phi _{I}\partial _{L}\Phi _{J}{{\,\mathrm{d}\,}}x^{K}\wedge *{{\,\mathrm{d}\,}}x^{L}\\&=\sum _{|K|=q}(-1)^{\sigma _{K}}\partial _{K}\Phi _{I}\partial _{K}\Phi _{J}{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}. \end{aligned}$$

Hence for

$$\begin{aligned}\omega =\sum _{I}\omega _{I}{{\,\mathrm{d}\,}}x^{I},\quad \Phi ^{*}\omega =\sum _{I}\tilde{\omega }_{I}\,\Phi ^{*}{{\,\mathrm{d}\,}}x^{I},\quad \tilde{\omega }:=\sum _{I}\tilde{\omega }_{I}{{\,\mathrm{d}\,}}x^{I},\qquad \tilde{\omega }_{I}:=\omega _{I}\circ \Phi \end{aligned}$$

we compute

$$\begin{aligned} *\,|\omega |^2 =\omega \wedge *\,\bar{\omega }&=\sum _{I,J}\omega _{I}\bar{\omega }_{J}{{\,\mathrm{d}\,}}x^{I}\wedge *{{\,\mathrm{d}\,}}x^{J} =\sum _{I}\omega _{I}\bar{\omega }_{I}{{\,\mathrm{d}\,}}x^{I}\wedge *{{\,\mathrm{d}\,}}x^{I}\\&=|\vec \omega |^2{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N},\\ *\,|\Phi ^{*}\omega |^2 =\Phi ^{*}\omega \wedge *\,\Phi ^{*}\bar{\omega }&=\sum _{I,J}\tilde{\omega }_{I}\bar{\tilde{\omega }}_{J}(\Phi ^{*}{{\,\mathrm{d}\,}}x^{I})\wedge *(\Phi ^{*}{{\,\mathrm{d}\,}}x^{J})\\&=\sum _{I,J}\sum _{|K|=q}(-1)^{\sigma _{K}}\tilde{\omega }_{I}\bar{\tilde{\omega }}_{J} \partial _{K}\Phi _{I}\partial _{K}\Phi _{J}{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}, \end{aligned}$$

and thus

$$\begin{aligned} |\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2&=|\omega |_{\mathsf {L}^{2,q}(\Omega )}^2 =\int _{\Omega }*\,|\omega |^2 =\int _{\Omega }|\vec \omega |^2{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}\\&=\int _{\Xi }|\vec {\tilde{\omega }}|^2\Phi ^{*}({{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N})\\&=\int _{\Xi }\det (\nabla \Phi )|\vec {\tilde{\omega }}|^2{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N} =\int _{\Xi }\det (\nabla \Phi )*|\tilde{\omega }|^2 =\int _{\Xi }\det (\nabla \Phi )|\vec {\tilde{\omega }}|^2,\\ |\overrightarrow{\Phi ^{*}\omega }|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2&=|\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2 =\int _{\Xi }*\,|\Phi ^{*}\omega |^2\\&=\sum _{I,J}\sum _{|K|=q}(-1)^{\sigma _{K}} \int _{\Xi }\tilde{\omega }_{I}\bar{\tilde{\omega }}_{J} \partial _{K}\Phi _{I}\partial _{K}\Phi _{J}{{\,\mathrm{d}\,}}x^{1}\wedge \dots \wedge {{\,\mathrm{d}\,}}x^{N}\\&=\sum _{I,J}\sum _{|K|=q}(-1)^{\sigma _{K}} \int _{\Xi }\tilde{\omega }_{I}\bar{\tilde{\omega }}_{J}\partial _{K}\Phi _{I}\partial _{K}\Phi _{J}. \end{aligned}$$

Therefore, we get

$$\begin{aligned} \min _{\Xi }\det (\nabla \Phi )\, |\tilde{\omega }|_{\mathsf {L}^{2,q}(\Xi )}^2 \le |\omega |_{\mathsf {L}^{2,q}(\Omega )}^2&\le \max _{\Xi }\det (\nabla \Phi )\, |\tilde{\omega }|_{\mathsf {L}^{2,q}(\Xi )}^2,\\ |\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2&\le N^{q}\left( {\begin{array}{c}N\\ q\end{array}}\right) ^{2} \max _{\Xi }|\nabla \Phi |^{2q}\, |\tilde{\omega }|_{\mathsf {L}^{2,q}(\Xi )}^2, \end{aligned}$$

where the second estimate is quite rough. Combing both we see

$$\begin{aligned} |\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2&\le c_{q,N,\nabla \Phi }|\omega |_{\mathsf {L}^{2,q}(\Omega )}^2,&c_{q,N,\nabla \Phi }&:=N^{q}\left( {\begin{array}{c}N\\ q\end{array}}\right) ^{2}\frac{\max _{\Xi }|\nabla \Phi |^{2q}}{\min _{\Xi }\det (\nabla \Phi )}, \end{aligned}$$
(B.1)
$$\begin{aligned} |\Psi ^{*}\zeta |_{\mathsf {L}^{2,q}(\Omega )}^2&\le c_{q,N,\nabla \Psi }|\zeta |_{\mathsf {L}^{2,q}(\Xi )}^2,&c_{q,N,\nabla \Psi }&:=N^{q}\left( {\begin{array}{c}N\\ q\end{array}}\right) ^{2}\frac{\max _{\Omega }|\nabla \Psi |^{2q}}{\min _{\Omega }\det (\nabla \Psi )} \end{aligned}$$
(B.2)

and with \(\omega =\Psi ^{*}\Phi ^{*}\omega \)

$$\begin{aligned} |\omega |_{\mathsf {L}^{2,q}(\Omega )}^2 \le c_{q,N,\nabla \Psi } |\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2,\qquad |\zeta |_{\mathsf {L}^{2,q}(\Xi )}^2 \le c_{q,N,\nabla \Phi }|\Psi ^{*}\zeta |_{\mathsf {L}^{2,q}(\Omega )}^2. \end{aligned}$$

Now we calculate by Theorem 3.6

$$\begin{aligned} \begin{aligned} |\omega |_{\mathsf {L}^{2,q}(\Omega )}^2&\le c_{q,N,\nabla \Psi }|\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2 \le c_{q,N,\nabla \Psi }c_{\mathsf {p},\Xi }^2\,\hat{\mu }^2 \big (|{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\Phi ^{*}\omega |_{\mathsf {L}^{2,q+1}(\Xi )}^2 +|\delta \mu \Phi ^{*}\omega |_{\mathsf {L}^{2,q-1}(\Xi )}^2\big )\\&=c_{q,N,\nabla \Psi }c_{\mathsf {p},\Xi }^2\,\hat{\mu }^2 \big (|\Phi ^{*}{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Xi )}^2 +|\Phi ^{*}*\delta \epsilon \,\omega |_{\mathsf {L}^{2,N-q+1}(\Xi )}^2\big )\\&\le c_{q,N,\nabla \Psi }c_{\mathsf {p},\Xi }^2\,\hat{\mu }^2 \big (c_{q+1,N,\nabla \Phi }|{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega )}^2 +c_{N-q+1,N,\nabla \Phi }|\delta \epsilon \,\omega |_{\mathsf {L}^{2,q-1}(\Omega )}^2\big )\\&\le c_{q,N,\nabla \Psi }\max \{c_{q+1,N,\nabla \Phi },c_{N-q+1,N,\nabla \Phi }\}c_{\mathsf {p},\Xi }^2\,\hat{\mu }^2 \big (|{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega )}^2 +|\delta \epsilon \,\omega |_{\mathsf {L}^{2,q-1}(\Omega )}^2\big )\\&\le c_{N}^4c_{\nabla \Phi ,\nabla \Psi }^4\,\hat{\mu }^2c_{\mathsf {p},\Xi }^2 \big (|{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega )}^2 +|\delta \epsilon \,\omega |_{\mathsf {L}^{2,q-1}(\Omega )}^2\big ), \end{aligned} \end{aligned}$$
(B.3)

i.e.,

$$\begin{aligned}c_{\mathsf {t},q,\epsilon }\le c_{N}^2c_{\nabla \Phi ,\nabla \Psi }^2\,\hat{\mu }\,c_{\mathsf {p},\Xi },\end{aligned}$$

with very rough constants

$$\begin{aligned} c_{N}:=N^{\nicefrac {N}{2}}N!,\qquad c_{\nabla \Phi ,\nabla \Psi }:= \frac{\max \big [\max _{\Xi }|\nabla \Phi |,\max _{\Omega }|\nabla \Psi |,1\big ]^{N}}{\min \big [\min _{\Xi }\sqrt{\det (\nabla \Phi )},\min _{\Omega }\sqrt{\det (\nabla \Psi )},1\big ]}. \end{aligned}$$
(B.4)

So, it remains to estimate \(\hat{\mu }\). For this we estimate for \(\Phi ^{*}\omega \in \mathsf {L}^{2,q}(\Xi )\)

$$\begin{aligned} \langle \mu \,\Phi ^{*}\omega ,\Phi ^{*}\omega \rangle _{\mathsf {L}^{2,q}(\Xi )}&=\pm \langle *\,\Phi ^{*}*\epsilon \,\omega ,\Phi ^{*}\omega \rangle _{\mathsf {L}^{2,q}(\Xi )} =\pm \langle \Phi ^{*}*\epsilon \,\omega ,*\,\Phi ^{*}\omega \rangle _{\mathsf {L}^{2,N-q}(\Xi )}\\&=\pm \int _{\Xi }(\Phi ^{*}*\epsilon \,\omega )\wedge (\Phi ^{*}\bar{\omega })\\&=\pm \int _{\Omega }*\,\epsilon \,\omega \wedge \bar{\omega } =\langle \epsilon \,\omega ,\omega \rangle _{\mathsf {L}^{2,q}(\Omega )} \le \overline{\epsilon }^2|\omega |_{\mathsf {L}^{2,q}(\Omega )}^2\\&\le \overline{\epsilon }^2c_{q,N,\nabla \Psi }|\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2,\\ \langle \mu \,\Phi ^{*}\omega ,\Phi ^{*}\omega \rangle _{\mathsf {L}^{2,q}(\Xi )}&=\langle \epsilon \,\omega ,\omega \rangle _{\mathsf {L}^{2,q}(\Omega )} \ge \underline{\epsilon }^{-2}|\omega |_{\mathsf {L}^{2,q}(\Omega )}^2\\&\ge \frac{1}{\underline{\epsilon }^2c_{q,N,\nabla \Phi }}|\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )}^2, \end{aligned}$$

and observe

$$\begin{aligned}\hat{\mu } \le \max \{\overline{\epsilon }\sqrt{c_{q,N,\nabla \Psi }},\underline{\epsilon }\sqrt{c_{q,N,\nabla \Phi }}\} \le \hat{\epsilon }\max \{\sqrt{c_{q,N,\nabla \Psi }},\sqrt{c_{q,N,\nabla \Phi }}\} \le \hat{\epsilon }\,c_{N}\,c_{\nabla \Phi ,\nabla \Psi }.\end{aligned}$$

Finally, this shows

$$\begin{aligned}c_{\mathsf {t},q,\epsilon }\le c_{N}^3c_{\nabla \Phi ,\nabla \Psi }^3\,\hat{\epsilon }\,c_{\mathsf {p},\Xi }.\end{aligned}$$

B.1. Classical vector analysis

Some of the latter estimates are very rough. Let us take a closer look at the classical case of vector analysis, i.e., at the special case of \(N=3\) and \(q=1\). By (3.3), see also Appendix C for more details and a rigorous proof, we know that \(\omega \) in \(\mathsf {D}^{q}(\Omega )\) resp. \(\mathring{\mathsf {D}}^{q}(\Omega )\) implies \(\Phi ^{*}\omega \) in \(\mathsf {D}^{q}(\Xi )\) resp. \(\mathring{\mathsf {D}}^{q}(\Xi )\) with \({{\,\mathrm{d}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{d}\,}}\omega \). For \(N=3\) and \(q=1\) this means for the vector proxy field \(\vec \omega \in \mathring{\mathsf {H}}({{\,\mathrm{curl}\,}},\Omega )\cong \mathring{\mathsf {D}}^{1}(\Omega )\) that

$$\begin{aligned}\overrightarrow{\Phi ^{*}\omega }=\nabla \Phi \,\vec {\tilde{\omega }}\in \mathring{\mathsf {H}}({{\,\mathrm{curl}\,}},\Xi )\cong \mathring{\mathsf {D}}^{1}(\Xi )\end{aligned}$$

with

$$\begin{aligned} {{\,\mathrm{curl}\,}}(\nabla \Phi \,\vec {\tilde{\omega }}) =\overrightarrow{{{\,\mathrm{d}\,}}\Phi ^{*}\omega } =\overrightarrow{\Phi ^{*}{{\,\mathrm{d}\,}}\omega } ={{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }, \end{aligned}$$
(B.5)

where \({{\,\mathrm{adj}\,}}(A)\) denotes the adjunct matrix of \(A\in \mathbb {R}^{3\times 3}\). If A is invertible it holds \({{\,\mathrm{adj}\,}}(A)=(\det A)A^{-1}\). For \(q=N-1=2\) we have for the vector proxy field \(\vec \omega \in \mathsf {H}({{\,\mathrm{div}\,}},\Omega )\cong \mathsf {D}^{2}(\Omega )\) that

$$\begin{aligned}\overrightarrow{\Phi ^{*}\omega } ={{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\,\vec {\tilde{\omega }}\in \mathsf {H}({{\,\mathrm{div}\,}},\Xi )\cong \mathsf {D}^{2}(\Xi )\end{aligned}$$

with

$$\begin{aligned}{{\,\mathrm{div}\,}}\big ({{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\,\vec {\tilde{\omega }}\big ) =\overrightarrow{{{\,\mathrm{d}\,}}\Phi ^{*}\omega } =\overrightarrow{\Phi ^{*}{{\,\mathrm{d}\,}}\omega } =\det (\nabla \Phi )\widetilde{{{\,\mathrm{div}\,}}\vec \omega }.\end{aligned}$$

Thus for \(\vec \omega \in \mathring{\mathsf {H}}({{\,\mathrm{curl}\,}},\Omega )\cap \epsilon ^{-1}\mathsf {H}({{\,\mathrm{div}\,}},\Omega )\) we have

$$\begin{aligned}\nabla \Phi \,\vec {\tilde{\omega }}\in \mathring{\mathsf {H}}({{\,\mathrm{curl}\,}},\Xi )\cap \mu ^{-1}\mathsf {H}({{\,\mathrm{div}\,}},\Xi ),\qquad \mu :=\frac{1}{\det (\nabla \Phi )}{{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\,\tilde{\epsilon }\,{{\,\mathrm{adj}\,}}(\nabla \Phi ),\end{aligned}$$

with (B.5) and

$$\begin{aligned} {{\,\mathrm{div}\,}}(\mu \nabla \Phi \,\vec {\tilde{\omega }})&={{\,\mathrm{div}\,}}\big ({{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\,\tilde{\epsilon }\,\vec {\tilde{\omega }}\big ) =\det (\nabla \Phi )\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }. \end{aligned}$$

Now we can compute (B.3) more carefully by

$$\begin{aligned} |\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2&=\int _{\Omega }|\vec \omega |^2 =\int _{\Xi }\det (\nabla \Phi )|\vec {\tilde{\omega }}|^2 \le \int _{\Xi }\det (\nabla \Phi )\big |(\nabla \Phi )^{-1}\big |^2|\nabla \Phi \,\vec {\tilde{\omega }}|^2\nonumber \\&=\int _{\Xi }\frac{1}{\det (\nabla \Phi )}\big |{{\,\mathrm{adj}\,}}(\nabla \Phi )\big |^2|\nabla \Phi \,\vec {\tilde{\omega }}|^2 \le \hat{c}_{\nabla \Phi }^2|\nabla \Phi \,\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2\nonumber \\&\le \hat{c}_{\nabla \Phi }^2c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\big |{{\,\mathrm{curl}\,}}(\nabla \Phi \,\vec {\tilde{\omega }})\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2 +\big |{{\,\mathrm{div}\,}}(\mu \nabla \Phi \,\vec {\tilde{\omega }})\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2\bigg )\nonumber \\&=\hat{c}_{\nabla \Phi }^2c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\big |{{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2 +\big |\det (\nabla \Phi )\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2\bigg )\nonumber \\&=\hat{c}_{\nabla \Phi }^2c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\int _{\Xi }\big |{{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }\big |^2 +\int _{\Xi }\big |\det (\nabla \Phi )\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }\big |^2\bigg )\nonumber \\&\le \hat{c}_{\nabla \Phi }^2c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\hat{c}_{\nabla \Phi }^2\int _{\Xi }\det (\nabla \Phi )|\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }|^2 +c_{\det (\nabla \Phi )}^2\int _{\Xi }\det (\nabla \Phi )|\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }|^2\bigg )\nonumber \\&=\hat{c}_{\nabla \Phi }^2c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\hat{c}_{\nabla \Phi }^2|{{\,\mathrm{curl}\,}}\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2 +c_{\det (\nabla \Phi )}^2|{{\,\mathrm{div}\,}}\epsilon \,\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2\bigg ), \end{aligned}$$
(B.6)

where

$$\begin{aligned} c_{\det (\nabla \Phi )}&:=\max _{\Xi }\sqrt{\det (\nabla \Phi )},\\ \hat{c}_{\nabla \Phi }&:=\max _{\Xi }\frac{\big |{{\,\mathrm{adj}\,}}(\nabla \Phi )\big |}{\sqrt{\det (\nabla \Phi )}} =\max _{\Xi }\sqrt{\det (\nabla \Phi )}\big |(\nabla \Phi )^{-1}\big | \le c_{\det (\nabla \Phi )}\max _{\Xi }\big |(\nabla \Phi )^{-1}\big |. \end{aligned}$$

Therefore, we have

$$\begin{aligned}c_{\mathsf {m,t},\epsilon }\le \hat{c}_{\nabla \Phi } \max \{\hat{c}_{\nabla \Phi },c_{\det (\nabla \Phi )}\} c_{\mathsf {m,t},\mu ,\Xi },\qquad c_{\mathsf {m,t},\mu ,\Xi } \le \hat{\mu }\,c_{\mathsf {p},\Xi },\end{aligned}$$

and it remains to estimate \(\hat{\mu }\). For this we compute for \(\vec {\tilde{\omega }}\in \overset{}{\mathsf {L}}{}^{2}_{}(\Xi )\)

$$\begin{aligned} \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&=\int _{\Xi }\mu \,\vec {\tilde{\omega }}\cdot \vec {\bar{\tilde{\omega }}} =\int _{\Xi }\det (\nabla \Phi )\big ((\nabla \Phi )^{-\top }\tilde{\epsilon }\,(\nabla \Phi )^{-1}\vec {\tilde{\omega }}\big )\cdot \vec {\bar{\tilde{\omega }}}\\&=\int _{\Xi }\det (\nabla \Phi )\big (\tilde{\epsilon }\,(\nabla \Phi )^{-1}\vec {\tilde{\omega }}\big )\cdot (\nabla \Phi )^{-1}\vec {\bar{\tilde{\omega }}}\\&=\int _{\Omega }(\epsilon \nabla \Psi \,\vec \omega )\cdot \nabla \Psi \,\vec {\bar{\omega }} =\langle \epsilon \nabla \Psi \,\vec \omega ,\nabla \Psi \,\vec \omega \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )} \end{aligned}$$

and estimate

$$\begin{aligned} \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&\le \overline{\epsilon }^2|\nabla \Psi \,\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2 =\overline{\epsilon }^2\int _{\Omega }|\nabla \Psi \,\vec \omega |^2 =\overline{\epsilon }^2\int _{\Xi }\det (\nabla \Phi )|(\nabla \Phi )^{-1}\vec {\tilde{\omega }}|^2\\&\le \overline{\epsilon }^2\int _{\Xi }\det (\nabla \Phi )|(\nabla \Phi )^{-1}|^2|\vec {\tilde{\omega }}|^2 \le \overline{\epsilon }^2\hat{c}_{\nabla \Phi }^2\int _{\Xi }|\vec {\tilde{\omega }}|^2 =\overline{\epsilon }^2\hat{c}_{\nabla \Phi }^2|\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2,\\ \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&\ge \underline{\epsilon }^{-2}|\nabla \Psi \,\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2 =\underline{\epsilon }^{-2}\int _{\Xi }\det (\nabla \Phi )|(\nabla \Phi )^{-1}\vec {\tilde{\omega }}|^2\\&\ge \underline{\epsilon }^{-2}\int _{\Xi }\frac{\det (\nabla \Phi )}{|\nabla \Phi |^{2}}|\vec {\tilde{\omega }}|^2 \ge \underline{\epsilon }^{-2}\check{c}_{\nabla \Phi }^{-2}\int _{\Xi }|\vec {\tilde{\omega }}|^2 =\frac{1}{\underline{\epsilon }^2\check{c}_{\nabla \Phi }^2}|\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2, \end{aligned}$$

where

$$\begin{aligned} \check{c}_{\nabla \Phi }&:=\max _{\Xi }\frac{|\nabla \Phi |}{\sqrt{\det (\nabla \Phi )}} =\frac{1}{\min _{\Xi }\frac{\sqrt{\det (\nabla \Phi )}}{|\nabla \Phi |}}. \end{aligned}$$

Finally, we obtain

$$\begin{aligned}\hat{\mu } \le \max \{\overline{\epsilon }\,\hat{c}_{\nabla \Phi },\underline{\epsilon }\,\check{c}_{\nabla \Phi }\} \le \hat{\epsilon }\max \{\hat{c}_{\nabla \Phi },\check{c}_{\nabla \Phi }\}\end{aligned}$$

and hence

$$\begin{aligned} c_{\mathsf {m,t},\epsilon }&\le \hat{c}_{\nabla \Phi } \max \{\hat{c}_{\nabla \Phi },c_{\det (\nabla \Phi )}\} \max \{\hat{c}_{\nabla \Phi },\check{c}_{\nabla \Phi }\} \,\hat{\epsilon }\,c_{\mathsf {p},\Xi }\nonumber \\&\le \max \{\hat{c}_{\nabla \Phi },\check{c}_{\nabla \Phi },c_{\det (\nabla \Phi )}\}^3 \,\hat{\epsilon }\,c_{\mathsf {p},\Xi }. \end{aligned}$$
(B.7)

Especially for \(\Phi (x):=r\,x\) with \(r>0\) we have

$$\begin{aligned} |\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2&=\int _{\Omega }|\vec \omega |^2 =\int _{\Xi }\det (\nabla \Phi )|\vec {\tilde{\omega }}|^2 =r|\nabla \Phi \,\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2\\&\le rc_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\big |{{\,\mathrm{curl}\,}}(\nabla \Phi \,\vec {\tilde{\omega }})\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2 +\big |{{\,\mathrm{div}\,}}(\mu \nabla \Phi \,\vec {\tilde{\omega }})\big |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2\bigg )\\&=rc_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (\int _{\Xi }\big |{{\,\mathrm{{{\,\mathrm{adj}\,}}^{\top }}\,}}(\nabla \Phi )\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }\big |^2 +\int _{\Xi }\big |\det (\nabla \Phi )\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }\big |^2\bigg )\\&=rc_{\mathsf {m,t},\mu ,\Xi }^2 \bigg ( r\int _{\Xi }\det (\nabla \Phi )|\widetilde{{{\,\mathrm{curl}\,}}\vec \omega }|^2 +r^{3}\int _{\Xi }\det (\nabla \Phi )|\widetilde{{{\,\mathrm{div}\,}}\epsilon \,\vec \omega }|^2\bigg )\\&=r^{2}c_{\mathsf {m,t},\mu ,\Xi }^2 \bigg (|{{\,\mathrm{curl}\,}}\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2 +r^2|{{\,\mathrm{div}\,}}\epsilon \,\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2\bigg ) \end{aligned}$$

and

$$\begin{aligned} \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&=\int _{\Xi }\mu \,\vec {\tilde{\omega }}\cdot \vec {\bar{\tilde{\omega }}} =\int _{\Xi }\det (\nabla \Phi )\big ((\nabla \Phi )^{-\top }\tilde{\epsilon }\,(\nabla \Phi )^{-1}\vec {\tilde{\omega }}\big )\cdot \vec {\bar{\tilde{\omega }}}\\&=r^{-2}\int _{\Xi }\det (\nabla \Phi )(\tilde{\epsilon }\,\vec {\tilde{\omega }})\cdot \vec {\bar{\tilde{\omega }}}\\&=r^{-2}\int _{\Omega }(\epsilon \,\vec \omega )\cdot \,\vec {\bar{\omega }} =r^{-2}\langle \epsilon \,\vec \omega ,\,\vec \omega \rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )},\\ \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&\le r^{-2}\overline{\epsilon }^2|\vec \omega |_{\overset{}{\mathsf {L}}{}^{2}_{}(\Omega )}^2 =r^{-2}\overline{\epsilon }^2\int _{\Xi }\det (\nabla \Phi )|\vec {\tilde{\omega }}|^2 =r\overline{\epsilon }^2|\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2,\\ \langle \mu \,\vec {\tilde{\omega }},\vec {\tilde{\omega }}\rangle _{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}&\ge r\underline{\epsilon }^{-2}|\vec {\tilde{\omega }}|_{\overset{}{\mathsf {L}}{}^{2}_{}(\Xi )}^2, \end{aligned}$$

i.e., \(\displaystyle \hat{\mu } \le \max \{\sqrt{r}\overline{\epsilon },\underline{\epsilon }/\sqrt{r}\} \le \frac{\max \{r,1\}}{\sqrt{r}}\hat{\epsilon }\), which shows

$$\begin{aligned}c_{\mathsf {m,t},\epsilon }\le r\max \{1,r\} c_{\mathsf {m,t},\mu ,\Xi } \le r\max \{1,r\} \,\hat{\mu }\,c_{\mathsf {p},\Xi } \le \sqrt{r}\max \{1,r\}^2 \,\hat{\epsilon }\,c_{\mathsf {p},\Xi }.\end{aligned}$$

On the other hand, (B.7) gives with \(c_{\det (\nabla \Phi )}=r^{\nicefrac {3}{2}}\), \(\hat{c}_{\nabla \Phi }=\sqrt{3}r^{\nicefrac {1}{2}}\), \(\check{c}_{\nabla \Phi }=\sqrt{3}r^{-\nicefrac {1}{2}}\) the less sharp estimate

$$\begin{aligned}c_{\mathsf {m,t},\epsilon }\le 3\sqrt{3}r^{\nicefrac {3}{2}}\max \{1,r^2\}^3 \,\hat{\epsilon }\,c_{\mathsf {p},\Xi }.\end{aligned}$$

Appendix C: Proof of (3.3) in the Bi-Lipschitz case

C.1. Without boundary conditions

For this, let \(\omega =\sum _{I}\omega _{I}{{\,\mathrm{d}\,}}x^{I}\in \mathsf {D}^{q}(\Omega )\). We have to prove \(\Phi ^{*}\omega \in \mathsf {D}^{q}(\Xi )\) with \({{\,\mathrm{d}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{d}\,}}\omega \). Let us first assume \(\omega \in \mathring{\mathsf {C}}^{\infty ,q}(\mathbb {R}^{N})\), i.e., \(\omega _{I}\in \mathring{\mathsf {C}}^{\infty }(\mathbb {R}^{N})\) for all I. By Appendix B we have

$$\begin{aligned} {{\,\mathrm{d}\,}}\Phi _{j}&=\sum _{i}\partial _{i}\Phi _{j}{{\,\mathrm{d}\,}}x^{i}, \\ \Phi ^{*}\omega&=\sum _{I}\tilde{\omega }_{I}\Phi ^{*}{{\,\mathrm{d}\,}}x^{I} =\sum _{I}\tilde{\omega }_{I}({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}}),\\ {{\,\mathrm{d}\,}}\omega&=\sum _{I,j}\partial _{j}\omega _{I}({{\,\mathrm{d}\,}}x_{j})\wedge ({{\,\mathrm{d}\,}}x^{I}). \end{aligned}$$

By Rademacher’s theorem we know that \(\tilde{\omega }_{I}=\omega _{I}\circ \Phi \) and \(\Phi _{j}\) belong to \(\mathsf {C}^{0,1}(\Xi )\subset \overset{}{\mathsf {H}}{}^{1}_{}(\Xi )\) and that the chain rule holds, i.e., \(\partial _{i}\tilde{\omega }_{I}=\sum _{j}\widetilde{\partial _{j}\omega _{I}}\partial _{i}\Phi _{j}\). As \(\Phi _{j}\in \overset{}{\mathsf {H}}{}^{1}_{}(\Xi )\) we get \({{\,\mathrm{d}\,}}\Phi _{j}\in \mathsf {D}^{1}_{0}(\Xi )\) by

$$\begin{aligned}\langle {{\,\mathrm{d}\,}}\Phi _{j},\delta \varphi \rangle _{\mathsf {L}^{2,1}(\Xi )} =-\langle \Phi _{j},\delta \delta \varphi \rangle _{\mathsf {L}^{2,0}(\Xi )} =0\end{aligned}$$

for all \(\varphi \in \mathring{\mathsf {C}}^{\infty ,2}(\Xi )\). Thus by definition we see

$$\begin{aligned} {{\,\mathrm{d}\,}}\Phi ^{*}\omega&=\sum _{I}({{\,\mathrm{d}\,}}\tilde{\omega }_{I})\wedge ({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}})\\&=\sum _{I,i}\partial _{i}\tilde{\omega }_{I} ({{\,\mathrm{d}\,}}x^{i})\wedge ({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}})\\&=\sum _{I,i,j}\widetilde{\partial _{j}\omega _{I}}\partial _{i}\Phi _{j} ({{\,\mathrm{d}\,}}x^{i})\wedge ({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}})\\&=\sum _{I,j}\widetilde{\partial _{j}\omega _{I}} ({{\,\mathrm{d}\,}}\Phi _{j})\wedge ({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}}). \end{aligned}$$

On the other hand it holds

$$\begin{aligned} \Phi ^{*}{{\,\mathrm{d}\,}}\omega&=\sum _{I,j}\widetilde{\partial _{j}\omega _{I}}(\Phi ^{*}{{\,\mathrm{d}\,}}x_{j})\wedge (\Phi ^{*}{{\,\mathrm{d}\,}}x^{I}) =\sum _{I,j}\widetilde{\partial _{j}\omega _{I}}({{\,\mathrm{d}\,}}\Phi _{j})\wedge ({{\,\mathrm{d}\,}}\Phi _{i_{1}})\wedge \dots \wedge ({{\,\mathrm{d}\,}}\Phi _{i_{q}}). \end{aligned}$$

Therefore, \(\Phi ^{*}\omega \in \mathsf {D}^{q}(\Xi )\) with \({{\,\mathrm{d}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{d}\,}}\omega \). For general \(\omega \in \mathsf {D}^{q}(\Omega )\) we pick \(\phi \in \mathring{\mathsf {C}}^{\infty ,q+1}(\Xi )\). The first part of the proof (for \(\omega =*\,\phi \) and \(\Phi =\Psi \)) shows \(\Psi ^{*}*\,\phi \in \mathsf {D}^{N-q-1}(\Omega )\) with \({{\,\mathrm{d}\,}}\Psi ^{*}*\,\phi =\Psi ^{*}{{\,\mathrm{d}\,}}*\,\phi \). As \({{\,\mathrm{supp}\,}}*\,\Psi ^{*}*\,\phi \) is a compact subset of \(\Omega \), standard mollification yields a sequence \((\Phi _{n})\subset \mathring{\mathsf {C}}^{\infty ,q+1}(\Omega )\) with \(\Phi _{n}\rightarrow *\,\Psi ^{*}*\,\phi \) in \(\Delta ^{q+1}(\Omega )\). Then

$$\begin{aligned}&\langle \Phi ^{*}\omega ,\delta \phi \rangle _{\mathsf {L}^{2,q}(\Xi )}\\&\quad =\int _{\Xi }\Phi ^{*}\omega \wedge *\delta \phi =\pm \int _{\Xi }\Phi ^{*}\omega \wedge \Phi ^{*}\Psi ^{*}{{\,\mathrm{d}\,}}*\,\phi =\pm \int _{\Xi }\Phi ^{*}(\omega \wedge \Psi ^{*}{{\,\mathrm{d}\,}}*\,\phi )\\&\quad =\pm \int _{\Omega }\omega \wedge \Psi ^{*}{{\,\mathrm{d}\,}}*\,\phi =\pm \int _{\Omega }\omega \wedge {{\,\mathrm{d}\,}}\Psi ^{*}*\,\phi =\pm \langle \omega ,\delta *\Psi ^{*}*\,\phi \rangle _{\mathsf {L}^{2,q}(\Omega )}\\&\quad \uparrow \;\pm \langle \omega ,\delta \Phi _{n}\rangle _{\mathsf {L}^{2,q}(\Omega )} =\pm \langle {{\,\mathrm{d}\,}}\omega ,\Phi _{n}\rangle _{\mathsf {L}^{2,q+1}(\Omega )}\\&\quad \downarrow \;\pm \langle {{\,\mathrm{d}\,}}\omega ,*\Psi ^{*}*\phi \rangle _{\mathsf {L}^{2,q+1}(\Omega )} =\pm \int _{\Omega }{{\,\mathrm{d}\,}}\omega \wedge \Psi ^{*}*\,\phi \\&\quad =\pm \int _{\Xi }\Phi ^{*}({{\,\mathrm{d}\,}}\omega \wedge \Psi ^{*}*\,\phi ) =\pm \int _{\Xi }(\Phi ^{*}{{\,\mathrm{d}\,}}\omega )\wedge *\,\phi =-\langle \Phi ^{*}{{\,\mathrm{d}\,}}\omega ,\phi \rangle _{\mathsf {L}^{2,q+1}(\Xi )} \end{aligned}$$

and hence \(\Phi ^{*}\omega \in \mathsf {D}^{q}(\Xi )\) with \({{\,\mathrm{d}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{d}\,}}\omega \). Finally, for \(\omega \in \epsilon ^{-1}\Delta ^{q}(\Omega )\) we have \(\epsilon \,\omega \in \Delta ^{q}(\Omega )\) and \(*\,\epsilon \,\omega \in \mathsf {D}^{N-q}(\Omega )\). Therefore, \(\Phi ^{*}*\,\epsilon \,\omega \in \mathsf {D}^{N-q}(\Xi )\) and \({{\,\mathrm{d}\,}}\Phi ^{*}*\,\epsilon \,\omega =\Phi ^{*}{{\,\mathrm{d}\,}}*\,\epsilon \,\omega =\pm \Phi ^{*}*\delta \,\epsilon \,\omega \) by the latter considerations. Hence

$$\begin{aligned}*\,\Phi ^{*}*\delta \,\epsilon \,\omega =\pm *\,{{\,\mathrm{d}\,}}\Phi ^{*}*\,\epsilon \,\omega =\pm \delta (\underbrace{*\,\Phi ^{*}*\,\epsilon \,\Psi ^{*}}_{=\pm \mu })\,\Phi ^{*}\omega \end{aligned}$$

and \(\mu \,\Phi ^{*}\omega \in \Delta ^{q}(\Xi )\). By (B.1) we see

$$\begin{aligned}|\Phi ^{*}\omega |_{\mathsf {L}^{2,q}(\Xi )} \le c\,|\omega |_{\mathsf {L}^{2,q}(\Omega )},\qquad |{{\,\mathrm{d}\,}}\Phi ^{*}\omega |_{\mathsf {L}^{2,q+1}(\Xi )} =|\Phi ^{*}{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Xi )} \le c\,|{{\,\mathrm{d}\,}}\omega |_{\mathsf {L}^{2,q+1}(\Omega )}\end{aligned}$$

and

$$\begin{aligned}|\delta \mu \,\Phi ^{*}\omega |_{\mathsf {L}^{2,q-1}(\Xi )} =|{{\,\mathrm{d}\,}}\Phi ^{*}*\epsilon \,\omega |_{\mathsf {L}^{2,N-q+1}(\Xi )} \le c\,|{{\,\mathrm{d}\,}}*\epsilon \,\omega |_{\mathsf {L}^{2,N-q+1}(\Omega )} =c\,|\delta \epsilon \,\omega |_{\mathsf {L}^{2,q-1}(\Omega )}.\end{aligned}$$

C.2. With boundary conditions

Let \(\omega \in \mathring{\mathsf {D}}^{q}(\Omega )\) and \((\omega _{n})\subset \mathring{\mathsf {C}}^{\infty ,q}(\Omega )\) with \(\omega _{n}\rightarrow \omega \) in \(\mathsf {D}^{q}(\Omega )\). By Appendix C.1 we know \(\Phi ^{*}\omega ,\Phi ^{*}\omega _{n}\in \mathsf {D}^{q}(\Xi )\) with \({{\,\mathrm{d}\,}}\Phi ^{*}\omega _{n}=\Phi ^{*}{{\,\mathrm{d}\,}}\omega _{n}\) as well as \({{\,\mathrm{d}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{d}\,}}\omega \). Since \(\Phi ^{*}\omega _{n}=\sum _{I}\tilde{\omega }_{n,I}\Phi ^{*}{{\,\mathrm{d}\,}}x^{I}\) holds, \(\Phi ^{*}\omega _{n}\) has compact support in \(\Xi \). By standard mollification we see \(\Phi ^{*}\omega _{n}\in \mathring{\mathsf {D}}^{q}(\Xi )\). Moreover, \(\Phi ^{*}\omega _{n}\rightarrow \Phi ^{*}\omega \) in \(\mathsf {D}^{q}(\Xi )\) as \(\Phi ^{*}\omega _{n}\rightarrow \Phi ^{*}\omega \) in \(\mathsf {L}^{2,q}(\Xi )\) and

$$\begin{aligned}{{\,\mathrm{d}\,}}\Phi ^{*}\omega _{n}=\Phi ^{*}{{\,\mathrm{d}\,}}\omega _{n}\rightarrow \Phi ^{*}{{\,\mathrm{d}\,}}\omega ={{\,\mathrm{d}\,}}\Phi ^{*}\omega \end{aligned}$$

in \(\mathsf {L}^{2,q+1}(\Xi )\) by (B.1). Therefore \(\Phi ^{*}\omega \in \mathring{\mathsf {D}}^{q}(\Xi )\) with \({{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\Phi ^{*}\omega =\Phi ^{*}{{\,\mathrm{\mathring{{{\,\mathrm{d}\,}}}}\,}}\omega \).

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Pauly, D. On the Maxwell and Friedrichs/Poincaré constants in ND. Math. Z. 293, 957–987 (2019). https://doi.org/10.1007/s00209-018-2218-7

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Keywords

  • Maxwell’s equations
  • Maxwell constant
  • Second Maxwell eigenvalue
  • Electro statics
  • Magneto statics
  • Poincaré inequality
  • Friedrichs inequality
  • Poincaré constant
  • Friedrichs constant

Mathematics Subject Classification

  • 35A23
  • 35Q61
  • 35E10
  • 35F15
  • 35R45
  • 46E40
  • 53A45