SCHRIFTENREIHE DER FAKULTÄT FÜR MATHEMATIK. On Maxwell s and Poincaré s Constants. Dirk Pauly SM-UDE

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1 SCHIFTENEIHE DE FAKULTÄT FÜ MATHEMATIK On Maxwell s and Poincaré s Constants by Dirk Pauly SM-UDE

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3 On Maxwell s and Poincaré s Constants Dirk Pauly November 11, 2013 Dedicated to Sergey Igorevich epin on the occasion of his 60th birthday Abstract We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs and Poincaré 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. Key Words Maxwell s equations, Maxwell constant, second Maxwell eigenvalue, electro statics, magneto statics, Poincaré s inequality, Friedrichs inequality, Poincaré s constant, Friedrichs constant Contents 1 Introduction 1 2 Preliminaries 2 3 The Maxwell Estimates 5 A Appendix: The Maxwell Estimates in Two Dimensions 11 1 Introduction It is well known that, e.g., for bounded Lipschitz domains Ω 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 ( v 2 c 2 m rot v 2 + div v 2), (1.1) Ω Ω if in addition v is perpendicular to the so called Dirichlet or Neumann fields, i.e., v w = 0 w H(Ω), Ω 1

4 2 Dirk Pauly where H(Ω) = { H D (Ω) := {w L 2 (Ω) : rot w = 0, div w = 0, w t Γ = 0}, if v t Γ = 0, H N (Ω) := {w L 2 (Ω) : rot w = 0, div w = 0, w n Γ = 0}, if v n Γ = 0 holds. Here, c m is a positive constant independent of v, which will be called Maxwell constant. See, e.g., [20, 21, 13, 26]. 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 N, on N-dimensional iemannian manifolds, for differential forms or in the case of inhomogeneous media. See, e.g.,[10, 15, 17, 21, 22, 23, 26, 27]. 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 3 c p, c m c p diam(ω)/π (1.2) 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 u 2 c 2 p, u 2 resp. u 2 c 2 p u 2 Ω holds, if u Γ = 0 resp. Ω Ω Ω u = 0. While the result (1.2) is already well known in two dimensions, even for general Lipschitz domains Ω 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 [19], 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 applicational point of view most interesting inequality c m c p also for polyhedral domains in 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. Let us note that our methods are only based on elementary calculations. Ω 2 Preliminaries Throughout this paper let Ω 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

5 On Maxwell s and Poincaré s Constants 3 rot = and the divergence div = denoted by H 1 (Ω) := H(grad; Ω), D(Ω) := H(div; Ω), (Ω) := H(rot; Ω), H 1 (Ω) := H(grad; Ω) := C H 1 (Ω) (Ω), D(Ω) := H(div; Ω) := C D(Ω) (Ω), (Ω) := H(rot; Ω) := C (Ω) (Ω). In the latter three Hilbert spaces the classical homogeneous scalar, normal and tangential boundary traces are generalized, respectively. An index zero at the lower right corner of the latter spaces indicates a vanishing derivative, e.g., 0 (Ω) := {E (Ω) : rot E = 0}, D 0 (Ω) := {E D(Ω) : div E = 0}. Moreover, we introduce a symmetric, bounded (L ) and uniformly positive definite matrix field ε : Ω 3 3 and the spaces of (harmonic) Dirichlet and Neumann fields H D,ε (Ω) := 0 (Ω) ε 1 D 0 (Ω), H N,ε (Ω) := 0 (Ω) ε 1 D 0 (Ω). We will also use the weighted ε-l 2 (Ω)-scalar product, Ω,ε := ε, Ω and the corresponding induced weighted ε-l 2 (Ω)-norm Ω,ε :=, 1/2 Ω,ε. Moreover, ε denotes orthogonality with respect to the ε-l 2 (Ω)-scalar product. If we equip L 2 (Ω) with this weighted scalar product we write L 2 ε(ω). If ε equals the identity id, we skip it in our notations, e.g., we write := id and H D (Ω) := H D,id (Ω). By the assumptions on ε we have ε, ε > 0 E L 2 (Ω) ε 2 E 2 Ω εe, E Ω ε 2 E 2 Ω (2.1) and we note E 2 Ω,ε = εe, E Ω = ε 1/2 E 2 Ω as well as εe Ω = ε 1/2 E Ω,ε. Thus, for all E L 2 (Ω) ε 1 E Ω E Ω,ε ε E Ω, ε 1 E Ω,ε εe Ω ε E Ω,ε. (2.2) For later purposes let us also define ˆε := max{ε, ε}. We have the following compact embeddings: H 1 (Ω) H 1 (Ω) L 2 (Ω) (ellich s selection theorem) (2.3) (Ω) ε 1 D(Ω) L 2 (Ω) (tangential Maxwell compactness property) (2.4) (Ω) ε 1 D(Ω) L 2 (Ω) (normal Maxwell compactness property) (2.5) It is well known and easy to prove by standard indirect arguments that (2.3) implies the Poincaré estimates c p, > 0 u H 1 (Ω) u Ω c p, u Ω, (2.6) c p > 0 u H 1 (Ω) u Ω c p u Ω. (2.7)

6 4 Dirk Pauly Analogously, (2.4) implies dim H D,ε (Ω) <, since the unit ball in H D,ε (Ω) is compact, and the tangential Maxwell estimate, i.e., there exists c m,t,ε > 0 such that E (Ω) ε 1 D(Ω) (1 π D )E Ω,ε c m,t,ε ( rot E 2 Ω + div εe 2 Ω) 1/2, (2.8) where π D : L 2 ε(ω) H D,ε (Ω) denotes the ε-l 2 (Ω)-orthogonal projector onto Dirichlet fields. Similar results hold if one replaces the tangential or electric boundary condition by the normal or magnetic one. More precisely, (2.5) implies dim H N,ε (Ω) < and the corresponding normal Maxwell estimate, i.e., there exists c m,n,ε > 0 such that H (Ω) ε 1 D(Ω) H π N H Ω,ε c m,n,ε ( rot H 2 Ω + div εh 2 Ω) 1/2, (2.9) where π N : L 2 ε(ω) H N,ε (Ω) denotes the ε-l 2 (Ω)-orthogonal projector onto Neumann fields. We note that c 2 m,t,ε + 1 can also be seen as the norm of the inverse M 1 of the corresponding electro static Maxwell operator M : (Ω) ε 1 D(Ω) H D,ε (Ω) ε rot (Ω) L 2 (Ω) E (rot E, div εe). The analogous statement holds for c m,n,ε as well. The compact embeddings (2.3)-(2.5) hold for more general bounded domains with weaker regularity of the boundary Γ, such as domains with cone property, restricted cone property or just p-cusp-property. See, e.g., [1, 2, 20, 21, 22, 23, 24, 26, 27, 28, 13]. Note that the Maxwell compactness properties and hence the Maxwell estimates hold for mixed boundary conditions as well, see [10, 7, 9]. The boundedness of the underlying domain Ω is crucial, since one has to work in weighted Sobolev spaces in unbounded, like exterior domains, see [11, 12, 13, 14, 15, 17, 16, 20, 24]. 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 L 2 ε(ω) = H 1 (Ω) ε ε 1 D 0 (Ω) = 0 (Ω) ε ε 1 rot (Ω) = H 1 (Ω) ε H D,ε (Ω) ε ε 1 rot (Ω), L 2 ε(ω) = H 1 (Ω) ε ε 1 D 0 (Ω) = 0 (Ω) ε ε 1 rot (Ω) = H 1 (Ω) ε H N,ε (Ω) ε ε 1 rot (Ω), d D := dim H D,ε (Ω) is finite and independent of ε. In particular, d D depends just on the topology of Ω. More precisely, d D = β 2, the second Betti number of Ω. A similar result holds also for the Neumann fields, i.e., d N := dim H N,ε (Ω) = β 1.

7 On Maxwell s and Poincaré s Constants 5 where ε denotes the orthogonal sum with respect the latter scalar product, and note H 1 (Ω) = 0 (Ω) H D,ε (Ω) ε, ε 1 rot (Ω) = ε 1 D 0 (Ω) H D,ε (Ω) ε, H 1 (Ω) = 0 (Ω) H N,ε (Ω) ε, ε 1 rot (Ω) = ε 1 D 0 (Ω) H N,ε (Ω) ε. Moreover, with (Ω) := (Ω) rot (Ω) = (Ω) D 0 (Ω) H N (Ω), (Ω) := (Ω) rot (Ω) = (Ω) D 0 (Ω) H D (Ω) we see rot (Ω) = rot (Ω), rot (Ω) = rot (Ω). 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 E (Ω) ε 1 D(Ω) E Ω,ε c m,t,ε ( rot E 2 Ω + div εe 2 Ω) 1/2, (2.10) H (Ω) ε 1 D(Ω) H Ω,ε c m,n,ε ( rot H 2 Ω + div εh 2 Ω) 1/2 (2.11) 0 (Ω) = H 1 (Ω), 0 (Ω) = H 1 (Ω), D 0 (Ω) = rot (Ω), D 0 (Ω) = rot (Ω) as well as the simple Helmholtz decompositions L 2 ε(ω) = H 1 (Ω) ε ε 1 rot (Ω), L 2 ε(ω) = H 1 (Ω) ε ε 1 rot (Ω). (2.12) 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. 3 The Maxwell Estimates First, we have an estimate for irrotational fields, which is well known. Lemma 1 For all E H 1 (Ω) ε 1 D(Ω) and all H H 1 (Ω) ε 1 D(Ω) E Ω,ε εc p, div εe Ω, H Ω,ε εc p div εh Ω. Note that convex domains are always Lipschitz, see e.g. [8].

8 6 Dirk Pauly Proof Pick a scalar potential ϕ H 1 (Ω) with E = ϕ. Then, by (2.6) E 2 Ω,ε = εe, ϕ Ω = div εe, ϕ Ω div εe Ω ϕ Ω c p, div εe Ω ϕ Ω = c p, div εe Ω E Ω εc p, div εe Ω E Ω,ε. Let ϕ H 1 (Ω) with H = ϕ and ϕ. Since εh D(Ω) we obtain as before and by (2.7) H 2 Ω,ε = εh, ϕ Ω = div εh, ϕ Ω div εh Ω ϕ Ω c p div εh Ω ϕ Ω = c p div εh Ω H Ω εc p div εh Ω H Ω,ε, which finishes the proof. emark 2 Without any change, Lemma 1 extends to Lipschitz domains Ω N of arbitrary dimension. To get similar estimates for solenoidal vector fields we need a crucial lemma from [1, Theorem 2.17], see also [25, 8, 6, 4] for related partial results. Lemma 3 and Let Ω be convex and E (Ω) D(Ω) or E (Ω) D(Ω). Then E H 1 (Ω) E 2 Ω rot E 2 Ω + div E 2 Ω. (3.1) We note that for E H 1 (Ω) it is clear that for any domain Ω 3 E 2 Ω = rot E 2 Ω + div E 2 Ω (3.2) holds since = rot rot div. This formula is no longer valid if E has just the tangential or normal boundary condition but for convex domains the inequality (3.1) remains true. Lemma 4 Let Ω be convex. For all vector fields E (Ω) ε 1 rot (Ω) and all vector fields H (Ω) ε 1 rot (Ω) Proof E Ω,ε εc p rot E Ω, H Ω,ε εc p rot H Ω. Since εe rot (Ω) = rot (Ω) there exists a vector potential field Φ (Ω) with rot Φ = εe and Φ H 1 (Ω) by Lemma 3 since (Ω) = (Ω) D 0 (Ω). Moreover, Φ = rot Ψ can be represented by some Ψ (Ω). Hence, for any constant vector a 3 we have Φ, a Ω = rot Ψ, a Ω = 0. Thus, Φ belongs to H 1 (Ω) ( 3 ). Then, since E (Ω) and by Lemma 3 we get E 2 Ω,ε = E, εe Ω = E, rot Φ Ω = rot E, Φ Ω rot E Ω Φ Ω c p rot E Ω Φ Ω c p rot E Ω rot Φ Ω = c p rot E Ω εe Ω εc p rot E Ω E Ω,ε.

9 On Maxwell s and Poincaré s Constants 7 Since εh rot (Ω) there exists a vector potential Φ (Ω) with rot Φ = εh. Using the Helmholtz decomposition L 2 (Ω) = 0 (Ω) rot (Ω), we decompose (Ω) H = H 0 + H rot 0 (Ω) (Ω). Then, rot H rot = rot H and again by Lemma 3 we see H rot H 1 (Ω). Let a 3 such that H rot a H 1 (Ω) ( 3 ). Since Φ (Ω) and rot Φ, H 0 Ω = 0 = rot Φ, a Ω as well as by Lemma 3 we obtain H 2 Ω,ε = εh, H Ω = rot Φ, H Ω = rot Φ, H rot a Ω εh Ω H rot a Ω c p εh Ω H rot Ω εc p H Ω,ε rot H rot Ω = εc p H Ω,ε rot H Ω, completing the proof. emark 5 It is well known that Lemma 4 holds in two dimensions for any Lipschitz domain Ω 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. Theorem 6 Let Ω be convex. Then, for all vector fields E (Ω) ε 1 D(Ω) and all vector fields H (Ω) ε 1 D(Ω) E 2 Ω,ε ε 2 c 2 p,0 div εe 2 Ω + ε 2 c 2 p rot E 2 Ω, H 2 Ω,ε ε 2 c 2 p div εh 2 Ω + ε 2 c 2 p rot H 2 Ω. Thus, c m,t,ε max{εc p,, εc p } and c m,t,ε, c m,n,ε ˆεc p ˆε diam(ω)/π. Proof By the Helmholtz decomposition (2.12) we have (Ω) ε 1 D(Ω) E = E + E rot H 1 (Ω) ε ε 1 rot (Ω) with E H 1 (Ω) ε 1 D(Ω) and E rot (Ω) ε 1 rot (Ω) as well as div εe = div εe, rot E rot = rot E. By Lemma 1 and Lemma 4 and orthogonality we obtain Similarly we have E 2 Ω,ε = E 2 Ω,ε + E rot 2 Ω,ε ε 2 c 2 p, div εe 2 Ω + ε 2 c 2 p rot E 2 Ω. (Ω) ε 1 D(Ω) H = H + H rot H 1 (Ω) ε ε 1 rot (Ω)

10 8 Dirk Pauly with H H 1 (Ω) ε 1 D(Ω) and H rot (Ω) ε 1 rot (Ω) as well as div εh = div εh, rot H rot = rot H. By Lemma 1 and Lemma 4 H 2 Ω,ε = H 2 Ω,ε + H rot 2 Ω,ε ε 2 c 2 p div εh 2 Ω + ε 2 c 2 p rot H 2 Ω, which finishes the proof. In the case ε = id we can estimate the Maxwell constants also from below. Theorem 7 Let Ω be convex. If ε = id then c p,0 c m,t, c m,n c p diam(ω)/π. Proof By Theorem 6 we just have to show the lower bound. Looking at (2.10) and (2.11) we get 1 c 2 m,t = inf 0 E (Ω) D(Ω) rot E 2 Ω + div E 2 Ω, E 2 Ω 1 c 2 m,n = inf 0 E (Ω) D(Ω) rot E 2 Ω + div E 2 Ω. E 2 Ω Thus 1 c 2 m,t, 1 c 2 m,n inf 0 E H 1 (Ω) rot E 2 Ω + div E 2 Ω E 2 Ω = inf 0 E H 1 (Ω) E 2 Ω E 2 Ω = 1, c 2 p,0 completing the proof. emark 8 (i) It is unclear but most probable that c p,0 < c m,t < c m,n = c p holds. In a forthcoming publication [18] we will show more and sharper estimates on the Maxwell constants, showing additional and sharp relations between the Maxwell and the Poincaré/Friedrichs/Steklov constants. (ii) It is well known that c p, = 1/ λ 1, c p = 1/ µ 2 hold, where λ 1 is the first Dirichlet and µ 2 the second Neumann eigenvalue of the Laplacian. We note that by Theorem 7 we have given a new proof of the estimate 0 < µ 2 λ 1, which is of course well known, since even 0 < µ n+1 < λ n, n 1, holds, see, e.g., [5] and the literature cited there. Moreover, by Theorem 7 the eigenvalues of the different Maxwell operators lie between µ 2 and λ 1.

11 On Maxwell s and Poincaré s Constants 9 (iii) Our results extend also to all polyhedrons which allow the H 1 (Ω)-regularity of the Maxwell spaces (Ω) D(Ω) and (Ω) D(Ω) or to domains whose boundaries consist of combinations of convex boundary parts and polygonal parts which allow the H 1 (Ω)-regularity. Is is shown in [4, Theorem 4.1] that (3.1), even (3.2), still holds for all E H 1 (Ω) (Ω) or E H 1 (Ω) D(Ω) if Ω is a polyhedron. We note that even some non-convex polyhedrons 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. Acknowledgements The author is deeply indebted to Sergey epin not only for bringing his attention to the problem of the Maxwell constants in 3D. Moreover, the author wants to thank Sebastian Bauer und Karl-Josef Witsch for long term fruitful and deep discussions. eferences [1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in threedimensional non-smooth domains. Math. Methods Appl. Sci., 21(9): , [2] C. Amrouche, P.G. Ciarlet, and P. (Jr.) Ciarlet. Weak vector and scalar potentials. Applications to Poincaré s theorem and Korn s inequality in Sobolev spaces with negative exponents. Anal. Appl. (Singap.), 8(1):1 17, [3] M. Bebendorf. A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen, 22(4): , [4] M. Costabel. A coercive bilinear form for Maxwell s equations. J. Math. Anal. Appl., 157(2): , [5] N. Filonov. On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator. St. Petersburg Math. J., 16(2): , [6] V. Girault and P.-A. aviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer (Series in Computational Mathematics), Heidelberg, [7] V. Gol dshtein, I. Mitrea, and M. Mitrea. Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. J. Math. Sci. (N.Y.), 172(3): , [8] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman (Advanced Publishing Program), Boston, The crucial point is that the unit normal is piecewise constant.

12 10 Dirk Pauly [9] T. Jakab, I. Mitrea, and M. Mitrea. On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Indiana Univ. Math. J., 58(5): , [10] F. Jochmann. A compactness result for vector fields with divergence and curl in L q (Ω) involving mixed boundary conditions. Appl. Anal., 66: , [11] P. Kuhn and D. Pauly. egularity results for generalized electro-magnetic problems. Analysis (Munich), 30(3): , [12]. Leis. Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math. Z., 106: , [13]. Leis. Initial Boundary Value Problems in Mathematical Physics. Teubner, Stuttgart, [14] D. Pauly. Low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains. Adv. Math. Sci. Appl., 16(2): , [15] D. Pauly. Generalized electro-magneto statics in nonsmooth exterior domains. Analysis (Munich), 27(4): , [16] D. Pauly. Complete low frequency asymptotics for time-harmonic generalized Maxwell equations in nonsmooth exterior domains. Asymptot. Anal., 60(3-4): , [17] D. Pauly. Hodge-Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media. Math. Methods Appl. Sci., 31: , [18] D. Pauly. On Maxwell s constants in 3D. submitted, [19] L.E. Payne and H.F. Weinberger. An optimal Poincaré inequality for convex domains. Arch. ational Mech. Anal., 5: , [20]. Picard. andwertaufgaben der verallgemeinerten Potentialtheorie. Math. Methods Appl. Sci., 3: , [21]. Picard. On the boundary value problems of electro- and magnetostatics. Proc. oy. Soc. Edinburgh Sect. A, 92: , [22]. Picard. An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z., 187: , [23]. Picard. Some decomposition theorems and their applications to non-linear potential theory and Hodge theory. Math. Methods Appl. Sci., 12:35 53, [24]. Picard, N. Weck, and K.-J. Witsch. Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis (Munich), 21: , 2001.

13 On Maxwell s and Poincaré s Constants 11 [25] J. Saranen. On an inequality of Friedrichs. Math. Scand., 51(2): , [26] C. Weber. A local compactness theorem for Maxwell s equations. Math. Methods Appl. Sci., 2:12 25, [27] N. Weck. Maxwell s boundary value problems on iemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl., 46: , [28] K.-J. Witsch. A remark on a compactness result in electromagnetic theory. Math. Methods Appl. Sci., 16: , A Appendix: The Maxwell Estimates in Two Dimensions Finally, we want to note that similar but simpler results hold in two dimensions as well. More precisely, for N = 2 the Maxwell constants can be estimated by the Poincaré constants in any bounded Lipschitz domain Ω 2. Although this is quite well known, we present the results for convenience and completeness. As noted before, Lemma 1 holds in any dimension. In two dimensions the rotation rot differs from the divergence div just by a 90 -rotation given by [ ] 0 1 :=, 2 = id, = = The same holds for the co-gradient := rot (as formal adjoint) and the gradient. More precisely, for smooth functions u and smooth vector fields v we have [ ] 2 u rot v = div v = 1 v 2 2 v 1, u = u =, 1 u div v = rot v, u = u and thus also u = div u = div u = rot u. For the vector Laplacian we have v = rot div. Furthermore, v (Ω) v D(Ω), v (Ω) v D(Ω). The Helmholtz decompositions read L 2 (Ω) = H 1 (Ω) ε ε 1 D 0 (Ω) = 0 (Ω) ε ε 1 H 1 (Ω) = H 1 (Ω) ε H D,ε (Ω) ε ε 1 H 1 (Ω), L 2 (Ω) = H 1 (Ω) ε ε 1 D 0 (Ω) = 0 (Ω) ε ε 1 H 1 (Ω) = H 1 (Ω) ε H N,ε (Ω) ε ε 1 H 1 (Ω)

14 12 Dirk Pauly and we note H 1 (Ω) = 0 (Ω) H D,ε (Ω) ε, ε 1 H 1 (Ω) = ε 1 D 0 (Ω) H D,ε (Ω) ε, H 1 (Ω) = 0 (Ω) H N,ε (Ω) ε, ε 1 H 1 (Ω) = ε 1 D 0 (Ω) H N,ε (Ω) ε. We also need the matrix ε := ε, which fulfills the same estimates as ε, i.e., for all E L 2 (Ω) ε 2 E 2 Ω ε E, E Ω ε 2 E 2 Ω, since ε E, E Ω = εe, E Ω and E Ω = E Ω. But then the inverse ε 1 all E L 2 (Ω) which immediately follows by (2.2), i.e., ε 1 E, E Ω = ε 1/2 ε 2 E 2 Ω ε 1 E, E Ω ε 2 E 2 Ω, E 2 Ω { ε 2 ε ε 1/2 E, ε 1/2 E Ω = ε 2 E 2 Ω ε 2 ε ε 1/2 E, ε 1/2 E Ω = ε 2 E 2 Ω. satisfies for Hence, for the inverse matrix ε 1 = ε 1 simply ε and ε has to be exchanged. Furthermore, we have ε ±1/2 = ε ±1/2. For the solenoidal fields we have the following: Lemma 9 For all E (Ω) ε 1 H 1 (Ω) and all H (Ω) ε 1 H 1 (Ω) E Ω,ε εc p rot E Ω, H Ω,ε εc p, rot H Ω. Proof Since E D(Ω) and εe H 1 (Ω) we have εe H 1 (Ω) ε D(Ω). By Lemma 1 (interchanging ε and ε) we get E Ω,ε = ε 1/2 E Ω = ε 1/2 εe Ω = ε 1/2 εe Ω = εe Ω,ε 1 εc p div ε 1 εe Ω = εc p rot E Ω. Analogously, as H D(Ω) and εh H 1 (Ω) we have εh H 1 (Ω) ε D(Ω). Again by Lemma 1 (and again interchanging ε and ε) we get H Ω,ε = ε 1/2 H Ω = ε 1/2 εh Ω = ε 1/2 εh Ω = εh Ω,ε 1 εc p, div ε 1 εh Ω = εc p, rot H Ω, which completes the proof. Finally, the main result in proved as Theorem 6 and 7, but taking into account that there are now possibly Dirichlet and Neumann fields.

15 On Maxwell s and Poincaré s Constants 13 Theorem 10 For all E (Ω) ε 1 D(Ω) and all H (Ω) ε 1 D(Ω) E π D E 2 Ω,ε ε 2 c 2 p, div εe 2 Ω + ε 2 c 2 p rot E 2 Ω, H π N H 2 Ω,ε ε 2 c 2 p div εh 2 Ω + ε 2 c 2 p, rot H 2 Ω. Thus, c m,t,ε max{εc p,, εc p }, c m,n,ε max{εc p, εc p, } and If Ω is simply connected and ε = id then c m,t,ε, c m,n,ε ˆεc p. c p,0 c m,t, c m,n c p diam(ω)/π. Proof Using the Helmholtz decomposition we have (Ω) ε 1 D(Ω) H D,ε (Ω) ε E π D E = E + E H 1 (Ω) ε ε 1 H 1 (Ω) with E H 1 (Ω) ε 1 D(Ω) and E (Ω) ε 1 H 1 (Ω) as well as div εe = div εe, rot E = rot E. Thus, by Lemma 1 and Lemma 9 as well as orthogonality we obtain E π D E 2 Ω,ε = E 2 Ω,ε + E 2 Ω,ε ε 2 c 2 p, div εe 2 Ω + ε 2 c 2 p rot E 2 Ω. Analogously, we decompose (Ω) ε 1 D(Ω) H N,ε (Ω) ε H π N H = H + H H 1 (Ω) ε ε 1 H 1 (Ω) with H H 1 (Ω) ε 1 D(Ω) and H (Ω) ε 1 H 1 (Ω) as well as div εh = div εh, rot H = rot H. As before, by Lemma 1, Lemma 9 and orthogonality we see H π N H 2 Ω,ε = H 2 Ω,ε + H 2 Ω,ε ε 2 c 2 p div εh 2 Ω + ε 2 c 2 p, rot H Ω, yielding the assertion for the upper bounds. For the lower bounds, we assume that Ω is simply connected, which is in two dimensions equivalent to a connected boundary. Hence, there are no Dirichlet or Neumann fields. If moreover ε = id, we can use the same arguments as in the proof of Theorem 7, which completes the proof.

16 IN DE SCHIFTENEIHE DE FAKULTÄT FÜ MATHEMATIK ZULETZT ESCHIENENE BEITÄGE: Nr. 769: Mali, O., Muzalevskiy, A., Pauly, D.: Conforming and Non-Conforming Functional A Posteriori Error Estimates for Elliptic Boundary Value Problems in Exterior Domains: Theory and Numerical Tests, 2013 Nr. 770: Bauer, S., Neff, P., Pauly, D., Starke, G.: Dev-Div- and DevSym- DevCurl-Inequalities for Incompatible Square Tensor Fields with Mixed Boundary Conditions, 2013 Nr. 771: Pauly, D.: On the Maxwell Inequalities for Bounded and Convex Domains, 2013 Nr. 772: Pauly, D.: On Maxwell's and Poincaré's Constants, 2013