L p Theory for the div-curl System

Transcription

1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 6, HIKARI Ltd, L p Theory for the div-curl System Junichi Aramaki Division of Science, Faculty of Science and Engineering Tokyo Denki University Hatoyama-machi, Saitama, , Japan Copyright c 2014 Junichi Aramaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to solve the div-curl system in the framework of L p theory. We shall give some sufficient conditions for the existence of solutions, and give the estimates of the solutions by the given data. We can also obtain the uniqueness of solution in the particular case of the domain. Mathematics Subject Classification: 35A01, 35J56, 35F45 Keywords: div-curl system, existence of solutions, L p estimates 1 Introduction In this paper, we shall consider the following div-curl system. or curl u = B, div u = f in Ω, ν u = g on Ω, (1.1) curl u = B, div u = f in Ω, ν u = h on Ω (1.2) where Ω is a bounded domain in R 3 which may be a multi-connected and have holes satisfying some regularity conditions and ν is the unit outer normal vector field on the boundary Ω, and a vector field and a function B,f are given in Ω, and a vector field and a function h,g are given on Ω, respectively. We shall consider the existence of solution for the problem (1.1) or (1.2) and estimate of solution by the given data. In the framework of L 2 theory

2 260 Junichi Aramaki and C α Schauder theory, Pan [10] gave the existence and the estimate in L 2 theory and C α Schauder theory. About the estimates, there are many articles. For example, see Aramaki [3], Bates and Pan [4], Dautray and Lions [7] for L 2 estimate, and Bolik and Wahl [5] and Wahl [11] for C α Schauder estimate. On the contrary, in the framework of L p theory it seems that there are few literatures (cf. Amrouche and Seloula [1, 2]). Thus we shall give the existence and estimate of solutions for the problem (1.1) or (1.2) in the framework of L p theory. Main theorems will be given in section 3 (Theorem 3.3 and Theorem 3.5). 2 Preliminaries In this section, we shall give the preliminaries for the div-curl system in the next section. First, we assume the regularity and shape of the domain Ω. Next, we give the well definedness of the trace to the boundary of Ω for some fields, and then some estimates. Throughout this paper we assume the following (O1) and (O2) with respect to the regularity and shape of the domain Ω. (O1) Ω is a bounded domain in R 3 with C r,1 (r 1) boundary Γ = Ω of dimention 2 and Ω is locally situated on one side of Γ. Γ has a finite number of connected components Γ 1,...,Γ m+1 where m 0 and Γ m+1 denoting the boundary of the infinite connected component of R 3 \ Ω. (O2) There exist n manifolds of dimension 2 and of class C 4 denoted by Σ 1,...,Σ n (n 0) such that Σ i Σ j = for i jand non-tangential to Γ, and Ω :=Ω\ Σ, where Σ = n i=1σ i, is simply connected and pseudo-c 1,1. Here according to [1], it is said that the bounded open set D in R 3 is called pseudo-c 1,1 if for any point x on the boundary D there exists an integer r(x) equal to 1 or 2 and a neighborhood of V x of x such that D V x has r(x) connected components, each boundary being of C 1,1. For 1 p<, if we define H p 1 (Ω) = {u Lp (Ω, R 3 ); div u =0, curl u = 0 in Ω, ν u = 0 on Γ}, then dim H p 1(Ω, R 3 )=n which is called the first Betti number. We note that Ω is of class C 1,1. We consider the following problem: for every i =1,...n, Δp =0 p Σ k in Ω, = 0 on Γ, [ ] p [p] Σk = constant, = 0 for k =1,...,n Σ k p ds = δ ik (k =1,...,n)

3 L p theory for the div-curl system 261 where [ ] Σk is the jump of a function across Σ k and, Σk denotes the duality of W 1 1/q,q (Σ k ) and its dual space for any 1 <q<. Since Ω isofc 1,1 class, it follows from [1] and Girault and Raviart [8] that this problem has a unique solution p i W 2,q ( Ω) for any 1 <q<. Since p i W 1,q ( Ω, R 3 ) L q ( Ω, R 3 ), we can extend to a field on Ω as a field of L q (Ω, R 3 ) which is denoted by p i. Then { p i } j=1,...,n is a basis of H p 1(Ω). See [1, Corollary 4.1]. Next we define H p 2 (Ω) = {u Lp (Ω, R 3 ); div u =0, curl u = 0 in Ω, ν u = 0 on Γ}, then dim H p 2(Ω) = m which is called the second Betti number. For every j = 1,...,m, the following problem: Δq = 0 on Ω, q Γm+1 =0,q Γj = constant for j =1,...,m, q Γ j ds = δ q jk for k =1,...,m, ds = 1 Γ m+1 has a unique solution q j W 1,q (Ω) for any 1 <q<. Then { q j } j=1,...,m is a basis of H p 2 (Ω). See [1, Corollary 4.2]. If n = 0, we say that Ω is simply connected, and if m = 0 we say that Ω has no holes. Let 1 <p< and define a space H p (Ω, curl ) = {u L p (Ω, R 3 ); curl u L p (Ω, R 3 )}, then the space is a Banach space with respect to the norm and define a space u H p (Ω,curl ) = curl u L p (Ω) + u L p (Ω), H p (Ω, div ) = {u L p (Ω, R 3 ); div u L p (Ω)}, then the space is a Banach space with respect to the norm moreover if we define u H p (Ω,div ) = div u L p (Ω) + u L p (Ω), H p (Ω, curl, div ) = H p (Ω, curl ) H p (Ω, div ), the space is also a Banach space with respect to the norm u H p (Ω,curl,div ) = curl u L p (Ω) + div u L p (Ω) + u L p (Ω). We note that C (Ω, R 3 ) is dense in H p (Ω, curl ) or H p (Ω,,div ). Next we can take the trace to Γ of vector fields in H p (Ω, curl ) or H p (Ω, div ).

4 262 Junichi Aramaki Lemma 2.1. Let 1 <p<. Then the following holds. (i) The normal trace map C (Ω, R 3 ) u ν u Γ can be extended to a continuous linear operator from H p (Ω, div ) to W 1/p,p (Γ). (ii) The tangential trace map C (Ω, R 3 ) u u T = u (ν u)ν = (ν u) ν can be extended to a continuous linear operator from H p (Ω, curl ) to W 1/p,p (Γ, R 3 ). For the proof, see Chen and Pan [6]. Thus for u H p (Ω, div ), ν u, 1 Γj (j =1,...,m+ 1) are well defined where, denotes the duality of W 1/p,p (Γ j ) and W 1/p,p (Γ j ), p is the conjugate number of p that is determined by 1/p +1/p = 1. Similararly, for u H p (Ω, curl ), ν u, 1 Γj (j =1,...,m+ 1) are well defined Here we give the curl-free or divergence-free lifting. Lemma 2.2. Let k 0 be an integer and let 1 <p<. If Ω R 3 is a bounded domain with C k+1,1 boundary Γ and g W k+1 1/p,p (Γ), there exists u W k+1,p (Ω, R 3 ) such that curl u = 0 in Ω and ν u = g on Γ, and there exists a constant C = C(p, Ω) such that u W k+1,p (Ω) C g W (Γ). k+1 1/p,p Proof. Define f = 1 gds = constant where ds is the surface area of Γ, Ω Γ then we have We consider the equation f W k,p (Ω) = f L p (Ω) C(Ω) g L p (Γ). { Δφ = f in Ω, = g on Γ. φ Since Ω fdx = Γ gds, the equation has a unique solution φ W k+2,p (Ω) up to an additive constant, and φ W k+2,p (Ω) C( f W k,p (Ω) + g W k+1 1/p,p (Γ)) C 1 g W k+1 1/p,p (Γ). Define u = φ W k+1,p (Ω). Then curl u = 0 in Ω and ν u = ν φ = g on Ω, and u W k+1,p (Ω) = φ W k+1,p (Ω) C φ W k+2,p (Ω) C 1 g W k+1 1/p,p (Γ). Lemma 2.3. Let k 0 be an integer and let 1 <p<. If Ω R 3 is a bounded domain with C k+1,1 boundary Γ and h W k+1 1/p,p (Γ) satisfying ν h =0on Γ, then there exists u W k+1,p (Ω, R 3 ) such that div u =0in Ω and u T = h on Γ, and there exists a constant C = C(p, Ω) such that u W k+1,p (Ω) C h W k+1 1/p,p (Γ).

5 L p theory for the div-curl system 263 Proof. We can choose v W k+1,p (Ω, R 3 ) such that v = h on Γ and satisfies v W k+1,p (Ω) C(Ω) h W (Γ). We consider the equation k+1 1/p,p { Δϕ = div v in Ω, ϕ = 0 on Γ. Since div v W k,p (Ω), this equation has a unique solution ϕ W k+2,p (Ω) satisfying the estimate ϕ W k+2,p (Ω) C div v W k,p (Ω) C 1 v W k+1,p (Ω) C 2 h W k+1 1/p,p (Γ). If we define u = v ϕ W k+1,p (Ω), we have div u = div v Δϕ = 0 in Ω and u T = v T ( ϕ) T = h T = h on Γ and u W k+1,p (Ω) v W k+1,p (Ω) + ϕ W k+1,p (Ω) C h W k+1 1/p,p (Γ). Next proposition which seems to be new characterizes the simply connectedness of the domain Ω. Proposition 2.4. Let 1 <p<. Then the domain Ω is simply connected, that is to say, the first Betti number is equal to zero if and only if there exists a constant C = C(p, Ω) > 0 such that for all u W 1,p (Ω, R 3 ), u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.1) Moreover in this case, the following estimate holds. u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.2) Proof. According to [11, Theorem 3.2], the first Betti number is equal to zero if and only if there exists a constant C = C(p, Ω) > 0 such that for all u W 1,p (Ω, R 3 ) satisfying ν u = 0 on Γ, u L p (Ω) C( curl u L p (Ω) + div u L p (Ω)). (2.3) From this result, it is clear that (2.1) implies the simply connectedness. Thus it suffices to prove the necessity. For all u W 1,p (Ω, R 3 ), it follows from Lemma 2.2 that there exists v W 1,p (Ω, R 3 ) such that ν v = ν u on Γ, curl v = 0 in Ω, and v W 1,p (Ω) C ν u W (Γ). (2.4) 1 1/p,p If we put w = u v W 1,p (Ω, R 3 ), then ν w = 0 on Γ. Therefore from (2.3), using (2.4) we have w L p (Ω) C( curl w L p (Ω) + div w L p (Ω)) C( curl u L p (Ω) + curl v L p (Ω) + div u L p (Ω) + div v L p (Ω)) C( curl u L p (Ω) + div u L p (Ω) + ν u W (Γ)). 1 1/p,p

6 264 Junichi Aramaki From (2.4), we see that u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). Assume that (2.2) is fault. From (2.1) we have u W 1,p (Ω) C( u L p (Ω) + curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.5) If (2.2) is fault, then there exists a sequence {u j } W 1,p (Ω, R 3 ) such that u j L p (Ω) = 1, div u j 0inL p (Ω), curl u j 0 in L p (Ω, R 3 ) and ν u j 0in W 1 1/p,p (Γ). From (2.5), passing to a subsequence, we may assume that u j u weakly in W 1,p (Ω, R 3 ) and strongly in L p (Ω, R 3 ). Thus we get u L p (Ω) =1, curl u = 0, div u = 0 and ν u = 0 on Γ. Since Ω is simply connected, there exists a function φ W 2,p (Ω) such that u = φ in Ω. The function φ satisfies the equation { Δφ = div u = 0 in Ω, φ = ν u = 0 on Γ. Thus φ = constant, so u = φ = 0. This is a contradiction to u L p (Ω) =1. The following two propositions follow from [1, Corollary 3.5 and Corollary 5.3]. In the particular case p = 2, see [7]. Proposition 2.5. Let k 0 be an integer and Ω satisfies (O1)-(O2) with r k +1. If u L p (Ω, R 3 ) satisfies div u W k,p (Ω), curl u W k,p (Ω, R 3 ) and ν u W k+1 1/p,p (Γ), then we have u W k+1,p (Ω, R 3 ) and there exists a constant C = C(p, k, Ω) such that u W k+1,p (Ω) C( u L p (Ω) + div u W k,p (Ω) + curl u W k,p (Ω) + ν u W k+1 1/p,p (Γ). Proposition 2.6. Let k 0 be an integer and Ω satisfies (O1)-(O2) with r k +1. If u L p (Ω, R 3 ) satisfies div u W k,p (Ω), curl u W k,p (Ω, R 3 ) and ν u W k+1 1/p,p (Γ, R 3 ), then it follows that u W k+1,p (Ω, R 3 ) and there exists a constant C = C(p, k, Ω) such that u W k+1,p (Ω) C( u L p (Ω) + div u W k,p (Ω) + curl u W k,p (Ω) + ν u W k+1 1/p,p (Γ). Proposition 2.7. Assume that Ω is simply connected and let k 0 be an integer. Then there exists a constant C such that for every u W k+1,p (Ω, R 3 ), u L p (Ω) C( curl u W k,p (Ω) + div u W k,p (Ω) + ν u W k+1 1/p,p (Γ)).

7 L p theory for the div-curl system 265 Proof. Assume that the conclusion is false. Then there exists a sequence {u j } W k+1,p (Ω, R 3 ) such that u j L p (Ω) = 1, and curl u j 0inW k,p (Ω, R 3 ), div u j 0inW k,p (Ω) and ν u j 0inW k+1 1/p,p (Γ). By Proposition 2.5, we see that {u j } is bounded in W k+1,p (Ω, R 3 ). Passing to a subsequence, we may assume that u j u weakly in W k+1,p (Ω, R 3 ) and strongly in L p (Ω, R 3 ). Thus u W k+1,p (Ω, R 3 ) satisfies that curl u = 0, div u = 0 in Ω and ν u = 0 on Γ, and u L p (Ω) =1. Thusu H p 1 (Ω). Since Ω is simply connected, H p 1(Ω) = {0}. Therefore u = 0 in Ω. This contradicts to u L p (Ω) =1. Combined Proposition 2.5 with Proposition 2.7, we easily get the following. Corollary 2.8. Assume that Ω is simply connected and let k 0 be an integer. Then there exists a constant C such that for every u W k+1,p (Ω, R 3 ), u W k+1,p (Ω) C( curl u W k,p (Ω) + div u W k,p (Ω) + ν u W k+1 1/p,p (Γ)). Next proposition extends the result of [11, Theorem 3.1]. Proposition 2.9. Let 1 <p<. Then the domain Ω has no holes, that is to say, the second Betti number is equal to zero if and only if there exists a constant C = C(p, Ω) > 0 such that for all u W 1,p (Ω, R 3 ), u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.6) Moreover in this case, the following estimate holds. u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.7) Proof. According to [11, Theorem 3.1], the second Betti number is equal to zero if and only if there exists a constant C = C(p, Ω) > 0 such that for all u W 1,p (Ω, R 3 ) satisfying ν u = 0 on Γ, u L p (Ω) C( curl u L p (Ω) + div u L p (Ω)). (2.8) From this result, it is clear that (2.6) implies that Ω has no holes. Thus it suffices to prove the necessity. For all u W 1,p (Ω, R 3 ), it follows from Lemma 2.3 that there exists v W 1,p (Ω, R 3 ) such that ν v = ν u on Γ, div v =0 in Ω, and v W 1,p (Ω) C ν u W (Γ). (2.9) 1 1/p,p If we put w = u v W 1,p (Ω, R 3 ), then ν w = 0 on Γ. Therefore from (2.8), using (2.9) we have w L p (Ω) C( curl w L p (Ω) + div w L p (Ω)) C( curl u L p (Ω) + curl v L p (Ω) + div u L p (Ω) + div v L p (Ω)) C( curl u L p (Ω) + div u L p (Ω) + ν u W (Γ)). 1 1/p,p

8 266 Junichi Aramaki Thus we have u L p (Ω) C( curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). Assume that (2.7) is fault. From (2.6) we have u W 1,p (Ω) C( u L p (Ω) + curl u L p (Ω) + div u L p (Ω) + ν u W 1 1/p,p (Γ)). (2.10) If (2.7) is fault, then there exists a sequence {u j } W 1,p (Ω, R 3 ) such that u j L p (Ω) = 1, div u j 0inL p (Ω), curl u j 0 in L p (Ω, R 3 ) and ν u j 0inW 1 1/p,p (Γ). From (2.10), passing to a subsequence, we may assume that u j u weakly in W 1,p (Ω, R 3 ) and strongly in L p (Ω, R 3 ). Thus we get u L p (Ω) = 1, curl u = 0, div u = 0 and ν u = 0 on Γ. Since Ω has no holes, we see that H p (Ω) = {0}. Since u H p (Ω) = {0}, this is a contradiction to u L p (Ω) =1. Proposition Assume that Ω has no holes and let k 0 be an integer. Then there exists a constant C such that for every u W k+1,p (Ω, R 3 ), u L p (Ω) C( curl u W k,p (Ω) + div u W k,p (Ω) + ν u W k+1 1/p,p (Γ)). Proof. Assume that the conclusion is false. Then there exists a sequence {u j } W k+1,p (Ω, R 3 ) such that u j L p (Ω) = 1, curl u j 0inW k,p (Ω, R 3, div u j 0inW k,p (Ω) and ν u j 0inW k+1 1/p,p (Γ, R 3 ). By Proposition 2.6, we see that {u j } is bounded in W k+1,p (Ω, R 3 ). Passing to a subsequence, we may assume that u j u weakly in W k+1,p (Ω, R 3 ) and strongly in L p (Ω, R 3 ). Thus u W k+1,p (Ω, R 3 ) satisfies that curl u = 0, div u = 0 in Ω and ν u = 0 on Γ, and u L p (Ω) =1. Thusu H p 2(Ω). Since Ω has no holes, H p 2 (Ω) = {0}. Therefore u = 0 in Ω. This contradicts to u L p (Ω) =1. Combined Proposition 2.6 with Proposition 2.10, we easily get the following. Corollary Assume that Ω has no holes and let k 0 be an integer. Then there exists a constant C such that for any u W k+1,p (Ω, R 3 ), u W k+1,p (Ω) C( curl u W k,p (Ω) + div u W k,p (Ω) + ν u W k+1 1/p,p (Γ)). 3 The div-curl system In this section, we shall give some sufficient conditions in order to solve the div-curl system (1.1) or (1.2) and then we obtain the estimates of solutions by the given data.

9 L p theory for the div-curl system 267 Lemma 3.1. Assume that Ω R 3 satisfies (O1)-(O2). Then the set curl W 1,p (Ω, R 3 ) is closed subspace in L p (Ω, R 3 ) and curl W 1,p (Ω, R 3 )={u L p (Ω, R 3 ); div u =0in Ω, ν u, 1 Γj =0j =1, 2,...,m+1} where ν u, 1 Γj is the duality of W 1/p,p (Γ) W 1/p,p (Γ) which is well defined from Lemma 2.1. Proof. Let v W 1,p (Ω, R 3 ). The equation { Δφ = div v in Ω, φ = ν v on Ω has a unique (up to an additive constant) solution φ W 2,p (Ω). Then if we define w = v φ W 1,p (Ω, R 3 ), we see that div w = 0 in Ω, ν w = 0on Ω and curl w = curl v in Ω. Thus we see that curl W 1,p (Ω, R 3 ) = curl W 1,p n0 (Ω, div 0) where W 1,p n0 (Ω, div 0) = {w W 1,p (Ω, R 3 ); div w = 0 in Ω, ν w = 0 on Γ}. We can easily see that W 1,p n0 (Ω, div 0) is a Banach space with respect to the norm curl w L p (Ω) + w L p (Ω) which is equivalent to W 1,p norm. Thus we can apply the following Peetre lemma. Lemma 3.2. Let E 0,E 1,E 2 be Banach spaces and A 1 : E 0 E 2 be a continuous linear operator, and A 2 : E 0 E 2 be a continuous and compact linear operator satisfying x E0 A 1 x E1 + A 2 x E2 for all x E 0. Then KerA 1 is of finite dimensional subspace and ImA 1 is closed subspace in E 1. If we set E 0 = W 1,p n0 (Ω, div 0),E 1 = E 2 = L p (Ω, R 3 ) and A 1 = curl,a 2 = I d where I d is the identity operator, it is easily seen that the above lemma is applicable. Thus we can see that ImA 1 = curl W 1,p n0 (Ω, div 0) = curl W 1,p (Ω, R 3 ) is a closed subspace in L p (Ω, R 3 ). It follow from [1, Lemma 4.1] that u L p (Ω, R 3 ) satisfies div u = 0 in Ω and ν u, 1 Γj = 0 for j =1, 2,...,m+1 if and only if there exists v W 1,p (Ω, R 3 ) such that u = curl v in Ω. This completes the proof of Lemma 3.1. We are ready to state the main theorems in this paper.

10 268 Junichi Aramaki Theorem 3.3. Assume that k 0 is an integer and Ω satisfies (O1)-(O2) with r k +1. Let B W k,p (Ω, R 3 ),f W k,p (Ω) and g W k+1 1/p,p (Γ) satisfy the following conditions div B =0in Ω, ν B, 1 Γj =0for j =1,...,m+1, (3.1) and the compatibility condition: Ω fdx = Γ gds. (3.2) Then the system (1.1) has a solution u W k+1,p (Ω, R 3 ) and there exists a constant C 1 = C 1 (Ω,k,p) such that u W k+1,p (Ω) C 1 ( B W k,p (Ω)+ f W k,p (Ω)+ u L p (Ω)+ g W k+1 1/p,p (Γ)). (3.3) In particular, if Ω is simply connected, then the solution is unique and there exists a constant C 2 = C 2 (Ω,k,p) such that u W k+1,p (Ω) C 2 ( B W k,p (Ω) + f W k,p (Ω) + g W k+1 1/p,p (Γ)). (3.4) Proof. Since B satisfies (3.1), it follows from Lemma 3.1 that there exists A W 1,p (Ω, R 3 ) such that B = curl A. We consider the equation { Δφ = f div A in Ω, φ = g ν A on Γ. (3.5) Using (3.2) and the divergence theorem, we can see that (f div A)dx = (g ν A)dS. Ω Hence the problem (3.5) has a solution φ W 2,p (Ω) which is unique up to an additive constant. If we define u = A + φ W 1,p (Ω, R 3 ), then u satisfies that curl u = curl A = B W k,p (Ω, R 3 ), div u = div A +Δφ = f W k,p (Ω) and ν u = ν A + φ = g W k+1 1/p,p (Γ). Therefore from Proposition 2.5 we see that u W k+1,p (Ω, R 3 ) and the estimate (3.3) holds. When Ω is simply connected, if v W 1,p (Ω, R 3 ) satisfies that curl v = 0, div v = 0 in Ω and ν v = 0 on Γ, then v H p 1(Ω). Since Ω is simply connected, H p 1 (Ω) = {0}. Thus we have v = 0. From this, the uniqueness of solution follows. The estimate (3.4) follows from Corollary 2.8. Proposition 3.4. Let u L p (Ω, R 3 ). Then u satisfies that div u =0in Ω, ν u =0on Γ and ν u, 1 Σi =0for i =1, 2,...,n if and only if there exists a vector potential v W 1,p (Ω, R 3 ) such that u = curl v, div v =0in Ω, ν v = 0 on Γ and ν v, 1 Γj =0for j =1, 2,...,m. Moreover, v is unique and there exists a constant C>0such that v W 1,p (Ω) C u L p (Ω). Γ

11 L p theory for the div-curl system 269 The proof was given in [1, Theorem 4.3]. Theorem 3.5. Assume that k 0 is an integer and Ω satisfies (O1)-(O2) with r k +1.LetB W k,p (Ω, R 3 ),f W k,p (Ω) and h W k+1 1/p,p (Γ, R 3 ) satisfy the following conditions div B =0in Ω, ν h = 0, ν B = ν curl (h ν) on Γ, (3.6) and there exists H W 1,p (Ω, R 3 ) such that ν H = h on Γ and ν (curl H B), 1 Σi =0for i =1, 2,...,n. (3.7) Then the system (1.2) has a solution u W k+1,p (Ω, R 3 ) and there exists a constant C 3 = C 3 (Ω,k,p) such that u W k+1,p (Ω) C 3 ( B W k,p (Ω) + f W k,p (Ω) + u L p (Ω) + h W k+1 1/p,p (Γ)). (3.8) In particular, if Ω has no holes, then the solution is unique and there exists a constant C 4 = C 4 (Ω,k,p) such that u W k+1,p (Ω) C 4 ( B W k,p (Ω) + f W k,p (Ω) + h W k+1 1/p,p (Γ)). (3.9) Proof. First the equation { Δφ = f div H in Ω, φ = 0 on Γ has a unique solution φ W 2,p (Ω). Then we put u = v + H + φ and rewrite the equation (1.2) into the system with respect to v. Since curl v = curl u curl H = B curl H in Ω, div v = div u div H Δφ = f div H Δφ = 0 in Ω, ν v = ν u ν H ν φ = h h = 0 on Γ, we have the new system curl v = B curl H, div v = 0 in Ω, ν v = 0 on Γ. (3.10) From (3.6), we see that div (B curl H) = div B = 0 in Ω, and on Γ, ν (B curl H) = ν B ν curl H = ν B ν curl H T = ν B ν curl ((ν H) ν) = ν B ν curl (h ν) =0.

12 270 Junichi Aramaki Here we used the property ν curl H = ν curl H T (cf. Monneau [9]). Moreover, from (3.7), we have ν (B curl H), 1 Σi = 0 for i = 1, 2,....n. Thus it follows from Proposition 3.4 that there exists v W 1,p (Ω, R 3 ) such that curl v = B curl H, div v = 0 in Ω, ν v = 0 on Ω and ν v, 1 Γj =0 for j =1, 2,...,m. Thus v W 1,p (Ω, R 3 ) is a solution of (3.10). Therefore u = v + H + φ W 1,p (Ω, R 3 ) is a solution of (1.2). From the system (1.2) and Proposition 2.6, we have u W k,p (Ω, R 3 ) and the estimate (3.8) holds. When Ω has no holes, if curl w = 0, div w = 0 in Ω and ν w = 0 on Γ, then w H p 2(Ω). Since Ω has no holes, we see that H p 2(Ω) = {0}. Hence we see that w = 0. The estimate (3.9) follows from Corollary References [1] Amrouche, C. and Seloula, N. H., L p -theory for vector potentials and Sobolev s inequality for vector fields. Application to the Stokes equations with pressure boundary conditions, Math. Models and Methods in Appl. Sci., 23, (2013), [2] Amrouche, C. and Seloula, N. H., L p -theory for vector potentials and Sobolev s inequality for vector fields., C. R. Acad. Sci. Paris, Ser. 1., 349, (2011), [3] Aramaki, J., Variational problems associated with smectic-c to nematic or smectic-a to nematic phase transition in the theory of liquid crystals, Far East J. Math. Sci., 41, No. 2, (2010), [4] Bates, P. W. and Pan, X. B., Nucleation of instability of the Meissner state of 3-dimensional superconductors, Commun. Math. Phys., 276, (2007), [5] Bolik, J. and Wahl, W., Estimating u in terms of div u, curl u, either (ν, u) orν u and the topology, Math. Methods Appl. Sci., 20, (1997), [6] Chen, J. and Pan, X. -B., Functionals with operator curl in an extended magnetostatic Born-Infeld model, SIAM J. Math. Anal., 45, (2013), [7] Dautray, R. and Lions, J. L., Mathematical Analysis and Numerical Method for Science and Technology 3, Springer Verlag, New York, (1990). [8] Girault, V. and Raviart, P. A., Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, Heidelberg, New York, Tokyo, 1986.

13 L p theory for the div-curl system 271 [9] Monneau, R., Quasilinear elliptic system arising in a three-dimensional type II superconductor for infinite κ, Nonlinear Anal., 52, (2003), [10] Pan, X. -B., Nucleation of instability of Meissner state of superconductors and related mathematical problems, Trends of Partial Differential Equations, Int. Press, Somervill, MA, (2010), [11] Wahl, W., Estimating u by div u, curl u., Math. Methods Appl. Sci., 15, (1992), Received: January 15, 2014