Abstract
We complement the classic results of Chap. 2 in two directions. In the first part, we review some recent results on the traces of vector fields, and especially the tangential trace of electromagnetic-like fields. In the second part, we focus on the extraction of potentials for curl-free and/or divergence-free fields and consequences. In this chapter, Ω is an open subset of \(\mathbb {R}^3\) with boundary Γ.
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Notes
- 1.
This is a generalized orthogonality property, in the sense that, given f1, f2 ∈ H1∕2(Γ), there exist two sequences of elements of H1(Γ), respectively \((f_1^k)_k\) and \((f_2^k)_k\), such that \(f_i^k\rightarrow f_i\) in H1∕2(Γ) for i = 1, 2, and one has
- 2.
We assume here that ∂Γ′∩ ∂Γ′′≠∅. Indeed, it is simple to check that the preceding study carries over to the case when ∂Γ′∩ ∂Γ′′ = ∅.
- 3.
One has \(meas(\dot \varOmega )=meas(\varOmega )\).
- 4.
This (very technical) result is proven in two steps:
-
1.
One introduces a sequence of nested topologically trivial domains (Ω p ) p such that \(\overline {\varOmega _p}\subset \varOmega \) for all p, ∪ p Ω p = Ω. The curl-free field is regularized by convolution over \(\mathbb {R}^3\), so that its restriction belongs to C1(Ω p ), with vanishing curl in Ω p . One may then apply the classical Stokes’ Theorem in Ω p to this smooth field and write it as a gradient in that domain, with a scalar potential that belongs to C2(Ω p ).
-
2.
One then goes to the limit (p →∞) to derive the existence of a scalar potential in Ω that belongs to H1(Ω), with the help of the Lions’ Lemma (Theorem 2.1.34).
-
1.
- 5.
- 6.
Given ℓ, the problem is equivalent to the variational formulation
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} \mathit{Find}\ {q_\ell}\in H^1_{zmv}(\varOmega_\ell)\ \mathit{such \ that}\\ \displaystyle\forall q\in H^1_{zmv}(\varOmega_\ell),\ (\operatorname{\mathrm{\mathbf{grad}}} {q_\ell}|\operatorname{\mathrm{\mathbf{grad}}} q)=\langle {\boldsymbol{v}}\cdot\boldsymbol{n},q\rangle_{H^{1/2}(\varGamma_\ell)} \end{array} \right.. \end{aligned}$$This variational formulation is well-posed, cf. the Lax-Milgram Theorem 4.2.8 and the Poincaré-Wirtinger inequality of Theorem 2.1.37 in \(H^1_{zmv}(\varOmega _\ell )\). Due to the continuity of the trace mapping γ0 (Theorem 2.1.62), choosing q = q ℓ yields \(\| \operatorname {\mathrm {\mathbf {grad}}} {q_\ell }\|{ }_{\boldsymbol {L}^2(\varOmega _\ell )}\le C_\ell \,\|{\boldsymbol {v}}\cdot \boldsymbol {n}\|{ }_{H^{-1/2}(\varGamma _\ell )}\) with C ℓ > 0 independent of v. Finally, using the continuity of the normal trace mapping (Theorem 2.2.18), one gets the bound
$$\displaystyle \begin{aligned} \|\operatorname{\mathrm{\mathbf{grad}}} {q_\ell}\|{}_{\boldsymbol{L}^2(\varOmega_\ell)}\le C^{\prime}_\ell\,\|{\boldsymbol{v}}\|{}_{\boldsymbol{H}(\mathrm{div},\varOmega)}, \end{aligned}$$with \(C^{\prime }_\ell >0\) independent of v.
- 7.
- 8.
Noting that
is equal to the scalar product \((\cdot ,\cdot )_{\boldsymbol {X}_N(\varOmega )}\) on \(\boldsymbol {X}_N^\varGamma (\varOmega )\), well-posedness simply stems from the Riesz Theorem 4.2.1. - 9.
Due to the Lax-Milgram Theorem 4.2.8 and to the Poincaré-Wirtinger inequality of Theorem 2.1.37 in \(H^1_{zmv}(\varOmega )\), the variational formulation is well-posed. In addition, one can obviously add the case of constant test functions q = cst in the variational formulation: \(( \operatorname {\mathrm {\mathbf {grad}}} q_m^0| \operatorname {\mathrm {\mathbf {grad}}} cst)=0=(\boldsymbol {y}_m| \operatorname {\mathrm {\mathbf {grad}}} cst)\). It follows that all test functions q ∈ H1(Ω) can be used, and hence one finds that \(\varDelta q_m^0=\mathrm {div}\,\boldsymbol {y}_m\) in Ω and \(\partial _n q_m^0=\boldsymbol {y}_m\cdot \boldsymbol {n}=0\) on Γ.
- 10.
The form
is equal to the scalar product \((\cdot ,\cdot )_{\boldsymbol {X}_T(\varOmega )}\) on \(\boldsymbol {X}_T^\varSigma (\varOmega )\), so well-posedness stems from the Riesz Theorem 4.2.1. - 11.
The problem is equivalent to the variational formulation
$$\displaystyle \begin{aligned} \left\{ \begin{array}{l} \mathit{Find}\ q\in H^1_{zmv}(\varOmega)\ \mathit{such \ that}\\ \displaystyle\forall q'\in H^1_{zmv}(\varOmega),\ (\operatorname{\mathrm{\mathbf{grad}}} q|\operatorname{\mathrm{\mathbf{grad}}} q')=-(g|q') \end{array} \right.. \end{aligned}$$This variational formulation is well-posed, cf. the Lax-Milgram Theorem 4.2.8 and the Poincaré-Wirtinger inequality of Theorem 2.1.37 in \(H^1_{zmv}(\varOmega _\ell )\).
is equal to the scalar product 
is equal to the scalar product References
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Assous, F., Ciarlet, P., Labrunie, S. (2018). Complements of Applied Functional Analysis. In: Mathematical Foundations of Computational Electromagnetism. Applied Mathematical Sciences, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-70842-3_3
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