A remark on the regularity of solutions of Maxwell's equations on Lipshitz domains

Let u be a vector field on a bounded Lipschitz domain in R, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space H! on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L. Let ft C R be a bounded simply connected domain with connected Lipschitz boundary F. This means that F can be represented locally as the graph of a Lipschitz function. For properties of Lipschitz domains, see [7], [3], [2]. In particular, F has the strict cone property. We consider real vector fields u on ft satisfying in the distributional sense u 6 L(tt); div u e L(Q); curl u e L{Q) . (1) We denote the inner product in L(il) by (•,•). It is well known that functions u satisfying (1) have boundary values n x it and n u in the Sobolev space JH ~/(F) defined in the distributional sense by the natural extension of the Green formulas (curlu, v) — (w, curlt;) = (2) (div u, (p) + (w, grad ) = < n • w, tp > (3) for all v,ipe Here n denotes the exterior normal vector which exists almost everywhere on F, and < -, • > is the natural duality in H~(T) x H(T) extending the L(T) inner product. It is known that for smooth domains (e.g., F G C'), each one of the two boundary conditions n x u G # 1 / 2 (F) or n • u G # 1 / 2 (F) (4) implies u G H^), see [2] and, for the case of homogeneous boundary conditions, [6], where one finds also a counterexample for a nonsmooth domain. Such counterexamples are derived from nonsmooth weak solutions v G H{^1) of the Neumann problem (dn := n • grad denotes the normal derivative) Av = g e L(n); dnv = 0 on F (5) If u = grad v, then u satisfies (1) and nu = 0 on F, and u G i/($l) if and only if v G iJ 1 + (0) . For smooth or convex domains, one knows that v G H(£l). If fi has a nonconvex edge of opening angle a?r, a > 1, then, in general, the solution t> of (5) is not in i/(fi) for s = I / a , hence u £ H($l). This upper bound s for the smoothness of u can be arbitrary close to 1/2. Regularity theorems for (1), (4) have applications in the numerical approximation of the Stokes problem [2] and in the analysis of initial-boundary value problems for Maxwell's equations [6]. The compact embedding into L(Q,) of the space of solutions of the time-harmonic Maxwell equations is needed for the principle of limiting absorption. This compact embedding result was shown by Week [10] for a clctss of piecewise smooth domains and by Weber [9] and Picard [8] for general Lipschitz domains. In these proofs, no regularity result for the solution u was used or obtained. See Leis' book [6] for a discussion. In this note, we use the result by Dahlberg, Jerison, and Kenig [4], [5] on the H/ regularity for solutions of the Dirichlet and Neumann problems with L data in potential theory (see Lemma 1 below). Together with arguments similar to those described by Girault and Raviart [2], this yields u G Hl{Vt) (Theorem 2). The compact embedding in L is an obvious consequence of this regularity. If instead of Lemma 1, one uses only the more elementary tools from [1], one obtains H/~ regularity for solutions of the Dirichlet and Neumann problems in potential theory and, consequently u G if~(Jl) for any e > 0. This kind of regularity is also known for the case of an open manifold F (screen problem). It suffices, of course, for the compact embedding result. The proof of the following result can be found in [4]. Lemma 1. (Dahlberg-Jerison-Kenig) Let v G H^) satisfy Av = 0 in H. Then the two conditions (i) v\reH\T) and (ii) dnv\reL (T) are equivalent. They imply v G i y r r s n V 1.•<• . a . Remarks. a.) The first assertion in the Lemma goes back to Necas [7]. b.) There are accompanying norm estimates, viz. There exist constants Ci, C2, C3, independent of v such that Ci\\dnv\\L2(r) ^ ll^xgradt; | |L2 ( r ) < C2\\dnv\\L2ir) c.) The boundary values are attained in a stronger sense than the distributional sense (2), (3), namely pointwise almost everywhere in the sense of nontangential maximal functions in L(T). Theorem 2. Let u satisfy the conditions (1) in ft and either nxueL(T) (6) or n-u£ L(T). (7) ThenueH{Q). If (1) is satisfied, then the two conditions (6) and (7) are equivalent. Proof. The proof follows the lines of [2], It is presented in detail to make sure that it is valid for Lipschitz domains. Let / : = cuxlu G L(T). Then d i v / = 0 in ft. According to [2, Ch. I, Thm 3.4] there exists w G #(ft) with curl w = f , div w = 0 in ft. (8) The construction of w is as follows: Choose a ball O containing ft in its interior and solve in O \ ft the Neumann problem: x G H\O \ ft) with AX = 0 in O \ ft ; dnX = n / o n r ; dnX = 0 on 8O . (9) Note that n-f G H"^^) satisfies the solvability condition = 0 because Define f0 := / i n ft, f0 := gradx in <p\ft, f0 := 0 in R \O. Then f0 G L (R) has compact support and satisfies div f0 = 0 in R . Therefore / 0 = curl w; for some w G fi(R) with divtl; = 0 in R. One obtains w for example by convolution of /o with a fundamental solution of the Laplace operator in R and taking the curl. Thus (8) is satisfied. The function z := u— w satisfies zeL(n) and curl* = 0 in ft. (10) Since il is simply connected, there exists v G if (ft) with