On Maxwell's and Poincare's Constants

We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincar\'e's constants. In other words, the second Maxwell eigenvalues lie between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue.


Introduction
It is well known that, e.g., for bounded Lipschitz domains Ω ⊂ R 3 , a square integrable vector field v having square integrable divergence div v and square integrable rotation vector field rot v as well as vanishing tangential or normal component on the boundary Γ, i.e, v t | Γ = 0 resp. v n | Γ = 0, satisfies the Maxwell estimate holds. Here, c m is a positive constant independent of v, which will be called Maxwell constant. See, e.g., [19,20,13,25]. We note that (1.1) is valid in much more general situations modulo some more or less obvious modifications, such as for mixed boundary conditions, in unbounded (like exterior) domains, in domains Ω ⊂ R N , on N-dimensional Riemannian manifolds, for differential forms or in the case of inhomogeneous media. See, e.g., [10,15,17,20,21,22,25,26]. So far, to the best of the author's knowledge, general bounds for the Maxwell constants c m are unknown. On the other hand, at least estimates for c m from above are very important from the point of view of applications, such as preconditioning or a priori and a posteriori error estimation for numerical methods.
In this contribution we will prove that for bounded and convex domains Ω ⊂ R 3 holds true, where 0 < c p,• < c p are the Poincaré constants, such that for all square integrable functions u having square integrable gradient ∇u holds, if u| Γ = 0 resp. Ω u = 0. While the result (1.2) is already well known in two dimensions, even for general Lipschitz domains Ω ⊂ R 2 (except of the last inequality), it is new in three dimensions. We note that the last inequality in (1.2) has been proved in the famous paper of Payne and Weinberger [18], where also the optimality of the estimate was shown. A small mistake in this paper has been corrected later in [3]. We will prove the crucial and from the point of view of applications most interesting inequality c m ≤ c p also for polyhedral domains in R 3 , which might not be convex but still allow the H 1 (Ω)-regularity for solutions of Maxwell's equations. We will give a general result for non-smooth and inhomogeneous, anisotropic media as well, and even a refinement of (1.2). Let us note that our methods are only based on elementary calculations.

Preliminaries
Throughout this paper let Ω ⊂ R 3 be a bounded Lipschitz domain. Many of our results hold true under weaker assumptions on the regularity of the boundary Γ := ∂Ω. Essentially we need the compact embeddings (2.3)-(2.5) to hold. We will use the standard Lebesgue spaces L 2 (Ω) of square integrable functions or vector (or even tensor) fields equipped with the usual L 2 (Ω)-scalar product · , · Ω and L 2 (Ω)-norm | · | Ω . Moreover, we will work with the standard L 2 (Ω)-Sobolev spaces for the gradient grad = ∇, the rotation rot = ∇× and the divergence div = ∇· denoted by .
We have the following compact embeddings: holds, where λ 1 is the first Dirichlet and µ 2 the second Neumann eigenvalue of the Laplacian. We even have 0 < µ n+1 < λ n for all n ∈ N, see e.g. [5] and the literature cited there.
As always in the theory of Maxwell's equations, we need another crucial tool, the Helmholtz or Weyl decompositions of vector fields into irrotational and solenoidal vector fields. We have where ⊕ ε denotes the orthogonal sum with respect the latter scalar product, and note Note that all occurring spaces are closed subspaces of L 2 (Ω), which follows immediately by the estimates (2.6)-(2.9). More details about the Helmholtz decompositions can be found e.g. in [13].
If Ω is even convex † we have some simplifications due to the vanishing of Dirichlet and Neumann fields, i.e., H D,ε (Ω) = H N,ε (Ω) = {0}. Then (2.8) and (2.9) simplify to and we have as well as the simple Helmholtz decompositions The aim of this paper is to give a computable estimate for the two Maxwell constants c m,t,ε and c m,n,ε .

The Maxwell Estimates
First, we have an estimate for irrotational fields, which is well known.
Remark 2 Without any change, Lemma 1 extends to Lipschitz domains Ω ⊂ R N of arbitrary dimension.
To get similar estimates for solenoidal vector fields we need a crucial lemma from [1, Theorem 2.17], see also [24,8,6,4] for related partial results.

Remark 5
It is well known that Lemma 4 holds in two dimensions for any Lipschitz domain Ω ⊂ R 2 . This follows immediately from Lemma 1 if we take into account that in two dimensions the rotation rot is given by the divergence div after 90 • -rotation of the vector field to which it is applied. We refer to the appendix for details.
Proof By the Helmholtz decomposition (2.12) we have By Lemma 1 and Lemma 4 and orthogonality we obtain Similarly we have (Ω) as well as div εH ∇ = div εH, rot H rot = rot H.
By Lemma 1 and Lemma 4 which finishes the proof.
Remark 8 The latter proof shows that Theorem 7 extends to any Lipschitz domain Ω ⊂ R N of arbitrary dimension with the appropriate changes for the rotation operator.
Combining Theorems 6 and 7 we obtain: Let Ω be convex. Then c p,• ε 3 ≤ c m,t,ε ≤εc p , c p,• ε 3 < c p ε 3 ≤ c m,n,ε ≤εc p and hence c p,• ε 3 ≤ c m,t,ε , c m,n,ε ≤εc p ≤ε diam(Ω)/π. If additionally ε = id, then if Ω is a polyhedron ‡ . We note that even some non-convex polyhedra admit the H 1 (Ω)-regularity of the Maxwell spaces depending on the angle of the corners, which are not allowed to by too pointy.

Remark 11
(i) We conjecture c p,• < c m,t < c m,n = c p for convex Ω ⊂ R 3 .
(ii) We note that by Theorem 9 we have given a new proof of the estimate 0 < µ 2 ≤ λ 1 for convex Ω ⊂ R 3 . Moreover, the absolute values of the eigenvalues of the different Maxwell operators (tangential or normal boundary condition) lie between √ µ 2 and √ λ 1 .
Finally, we note that in the case ε = id we can find some different proofs for the lower bounds in less general settings. For example, if Ω has a connected boundary, then H D (Ω) = {0} and hence 1 c 2 constants in any bounded Lipschitz domain Ω ⊂ R 2 . Although this is quite well known, we present the results for convenience and completeness.
Finally, the main result is proved as Theorems 6, 7 and 9, but taking into account that there are now possibly Dirichlet and Neumann fields.