Quasiconvexity and Relaxation in Optimal Transportation of Closed Differential Forms

Abstract

This manuscript extends the relaxation theory from nonlinear elasticity to electromagnetism and to actions defined on paths of differential forms. The introduction of a gauge allows for a reformulation of the notion of quasiconvexity in Bandyopadhyay et al. (J Eur Math Soc 17:1009–1039, 2015), from the static to the dynamic case. These gauges drastically simplify our analysis. Any non-negative coercive Borel cost function admits a quasiconvex envelope for which a representation formula is provided. The action induced by the envelope not only has the same infimum as the original action, but has the virtue to admit minimizers. This completes our relaxation theory program.

Appendix: Systems of the Type \(\left( d,\delta \right) \) and Poincaré lemma

We start with a classical theorem which can be found, for instance, in [6] Theorem 7.2 or Schwarz [22].

Theorem 5.1

Let $1\le k\le n$ be an integer, $1<s<\infty$ and $\Omega \subset {\mathbb {R}}^{n}$ be a bounded open smooth contractible set with exterior unit normal $\nu .$ Then the following statements are equivalent:

• (i) $f\in L^{s}\left( \Omega ;\Lambda ^{k}\right) ,$$g\in L^{s}\left( \Omega ;\Lambda ^{k-2}\right)$ and $F_{0}\in W^{1,s}\left( \Omega ;\Lambda ^{k-1}\right)$ satisfy

  $$\begin{aligned} \left\{ \begin{array}[c]{cl} {\displaystyle \int _{\Omega }} \langle f;\delta \varphi \rangle - {\displaystyle \int _{\partial \Omega }} \langle \nu \wedge F_{0};\delta \varphi \rangle =0,\;\forall \,\varphi \in C^{\infty }\left( \overline{\Omega };\Lambda ^{k+1}\right) &{} \text {if }1\le k\le n-1\\ {\displaystyle \int _{\Omega }} f= {\displaystyle \int _{\partial \Omega }} \nu \wedge F_{0} &{} \text {if }k=n \end{array} \right. \end{aligned}$$
  $$\begin{aligned} {\displaystyle \int _{\Omega }} \langle g;\mathrm{d}\varphi \rangle =0,\;\forall \,\varphi \in C_{0}^{\infty }\left( \Omega ;\Lambda ^{k-3}\right) ; \end{aligned}$$

• (ii) There exists $F\in W^{1,s}(\Omega ;\Lambda ^{k-1})$ such that

  $$\begin{aligned} \left\{ \begin{array}[c]{cl} \mathrm{d}F=f\quad \text {and}\quad \delta F=g &{} \text {in }\Omega \\ \nu \wedge F=\nu \wedge F_{0} &{} \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

Remark 5.2

• (i) If $1\le k\le n-1,$ then the conditions in (i) just mean, in the weak sense, that

  $$\begin{aligned} \left[ \mathrm{d}F=0\text { and }\delta g=0\;\text {in }\Omega \right] \text { and }\left[ \nu \wedge f=\nu \wedge \mathrm{d}F_{0}\;\text {on }\partial \Omega \right] . \end{aligned}$$

• (ii) If $k=1,$ then the terms $\delta F$ and g are not present, while if $k=2,$ then $\delta g=0$ automatically.

The preceding theorem leads to the Poincaré lemma (cf., for example, Theorem 8.16 in [6]).

Theorem 5.3

(Poincaré Lemma). Let $1\le k\le n$ be an integer, $1<s<\infty$ and $\Omega \subset {\mathbb {R}}^{n}$ be a bounded open smooth contractible set with exterior unit normal $\nu .$ Then the following statements are equivalent:

• (i) $f\in L^{s}\left( \Omega ;\Lambda ^{k}\right)$ and $F_{0}\in W^{1,s}\left( \Omega ;\Lambda ^{k-1}\right)$ satisfy

  $$\begin{aligned} \left\{ \begin{array}[c]{cl} {\displaystyle \int _{\Omega }} \langle f;\delta \varphi \rangle - {\displaystyle \int _{\partial \Omega }} \langle \nu \wedge F_{0};\delta \varphi \rangle =0,\;\forall \,\varphi \in C^{\infty }\left( \overline{\Omega };\Lambda ^{k+1}\right) &{} \text {if }1\le k\le n-1\\ {\displaystyle \int _{\Omega }} f= {\displaystyle \int _{\partial \Omega }} \nu \wedge F_{0} &{} \text {if }k=n; \end{array} \right. \end{aligned}$$

• (ii) There exists $F\in W^{1,s}(\Omega ;\Lambda ^{k-1})\$such that

  $$\begin{aligned} \left\{ \begin{array}[c]{cl} \mathrm{d}F=f &{}\quad \text {in }\Omega \\ F=F_{0} &{}\quad \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$