The Rot-Div System in Exterior Domains

The goal of this paper is to reconsider the classical elliptic system rot v =  f, div v =  g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the Lp-framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for $${{\bf f} \in \dot B^s_{p,q}(\Omega)}$$f∈B˙p,qs(Ω) for $${-1+\frac 1p < s < \frac 1p}$$-1+1p<s<1p. As a limit case we prove the result for $${{\bf f} \in \dot B^0_{3,1}(\Omega)}$$f∈B˙3,10(Ω), provided the boundary is merely in $${B^{2-1/3}_{3,1}}$$B3,12-1/3. The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).


Introduction
The subject of the present paper is to analyze solutions to the following problem in a three dimensional domain Ω, where v is a sought vector field, f , g, b are given data and rot, div are the vorticity and divergence operators, n is the outer normal to ∂Ω. Due to questions concerning the kernel of operator (1.1) we restrict ourself to simply connected domains with bounded connected boundaries. Thus we obtain two classes: bounded and exterior domains. We assume that all functions have a certain decay at infinity expressed by the corresponding spaces the functions belong to. Note that the vorticity operator is sometimes denoted by curl, however, we prefer the notation rot.
The above problem is one of the most fundamental linear systems in physics, we find it in electromagnetism (the Maxwell equations) or in fluid mechanics [14][15][16][17][18]. In case of incompressible fluids (div v = 0) it enables to recover the velocity from the knowledge of the vorticity. In case of electromagnetism, having electric field, we are able to recover information about the magnetic field and vice-versa. It is closely related to the Helmholtz decomposition of vector fields [14]. Its two dimensional version leads to a trivial system being just a scalar Dirichlet problem for the Laplace operator. More than the three dimensional versions have no reasonable physical meaning. Thus we concentrate here our attention on the most interesting case, i.e. on the three dimensional one. From the point of view of the theory of PDEs, equations (1.1) are a nontrivial version of elliptic system of order one. Additionally, the exterior domain gets us closer to issues arising from application problems like description of flows past an obstacle.
System (1.1) can be simplified. We solve the following Neumann problem Δp = g in Ω,  We are required to assume the following compatibility condition Ω g dx = ∂Ω b dσ at least in a weak sense, provided the integrals exist. Subtracting the gradient of a solution to (1.2) from a solution to (1.1), and substituting we arrive in (1.1) to the following canonical form rot v = f in Ω, div v = 0 in Ω, n · v = 0 on ∂Ω. (1.4) The compatibility condition to system (1.4) says that div f = 0, since div rot ≡ 0.
The main objective of the present paper is the issue of existence and uniqueness to system (1.4). We consider our problem in the setting of Sobolev-Besov spaces. The first result reads (1. 6) We claim that v = 0. Since the first group of homology π 1 (Ω) is trivial -the domain is simply connected, so the Poincaré lemma (see e.g. [20]) says (we use the fact that ∂Ω is connected and bounded and Ω ⊂ R 3 ) that if rot v = 0 then there exists a scalar function h with ∇h ∈ W 1 p (Ω; R 3 ) such that v = ∇h. Then (1.6) reads Δh = 0 in Ω, n · ∇h = 0 on ∂Ω, (1.7) so the maximum principle yields ∇h = 0, i.e. v ≡ 0. Similar arguments yield the uniqueness also in the subsequent theorems.
in Ω and f · n = 0 at ∂Ω. Then there exists unique solution v to (1.4) with ∇v ∈Ḃ s p,q (Ω; R 3×3 ) and ∇v Ḃs p,q (Ω) ≤ C f Ḃs p,q (Ω) . (1.8) The limit case of the above theorem, for the optimal regularity of the boundary in the critical Besov spaces, is the following. (1.9) Note that in situations considered in Theorems 1.3 and 1.4 the normal component of the trace of f is well defined, see [8]. The assumption put on the domain implies that we consider only bounded sets being diffeomorphic with a ball or exterior domains such that R 3 \Ω is a set diffeomorphic with a ball. Such restriction guarantees the uniqueness of considered problems in the proofs. Throughout the whole paper we consider only domains with boundary regularity at least C 1,0 .
To describe the mathematical motivation of our paper, we first note that this important system has not been studied so intensively in the mathematical literature. The classical results are due to Friedrichs [13] and Solonnikov [26], some extensions of these results one can find in [2,9,24,31]. Note that some of the current results for this system concern numerical analysis [9,24]. The papers [1] and [30] study the problem in the L p framework ( [1] also in the W k,p framework) also in domains with non-zero Betti's number. However, the authors assume that the boundary is sufficiently smooth. They also considered several types of boundary conditions. In [6], the result was extended to Hölder spaces. All three papers were based on estimates of corresponding potentials.
Here we want to reconsider the proof, giving some extension for solvability in the Besov class. Our method is based on the a priori bounds and compactness, omitting technical theory of potentials. Additionally, we examine rigorously the case of exterior domain. As the main new result we obtain the existence in the limit Besov spaces with the lowest regularity of the boundary (see Theorem 1.4), such results are necessary in order to consider free boundary problem in different areas of fluid mechanics and electromagnetism.
Let us discuss the chosen regularity. We consider regular solutions, i.e., the gradient of the solution is integrable, or at least almost integrable. We shall look closer at the choice of regularity of the boundary which is a key element of our considerations. In Theorem 1.1 it is required that ∂Ω ∈ W 2−1/q q with q > 3. It follows that the normal vector field is at least Hölder continuous C 0,α with α = 1 − 3/p. Next, Theorem 1.3 extends the results for Sobolev spaces on the Besov setting, we are interested only in the case as s is in a neighborhood of zero, since such considerations are related to the issue of optimal regularity. One can find this restriction in current works concerning free boundary problems in the optimal regularity [10][11][12][21][22][23]27,28,32]. Theorem 1.4 reaches the limit case, namely, ∂Ω ∈ B 5/3 3,1 , what implies that the normal vector field is C ω(·) with some modulus of continuity ω(·).
The paper is organized as follows. First we reformulate the problem into the second order elliptic equations and present the necessary definitions and few elementary facts for the Laplace operator and for products in Besov spaces. In Sect. 3 we prove the elementary existence of L 2 solutions taking care of the regularity of the boundary. Next, we consider the L p case, showing that the weak solution can be of better regularity. Hence we obtain the result in the Sobolev spaces. In Remark 4.4 we explain why the proof of Theorem 1.3 is not given. See also Remark 3.5. At the end we investigate the limit case, i.e., we prove Theorem 1.4.
Remark 2.2. Indeed, condition ∂Ω e · n dσ = 0 is irrelevant in order to show that solutions to (2.1) generate solutions to (1.4). On the other hand, this condition allows us to get unique solutions to (2.1) which is useful in order to get estimates of solutions to (2.1) in different spaces and thus construct solutions to our main problem.
Note that the condition ∂Ω e · n dσ = 0 for a solution to (2.1) is automatically fulfilled in the case of a bounded domain with connected boundary; it is a consequence of the global constraint div e = 0 coming from the fact that div f = 0, and the boundary condition (2.1) 3 . But in the case of exterior problem or in the case when the boundary consists of several connected parts we are able to obtain a nontrivial kernel of the operator without this integral constraint. Since ∂Ω is connected in our case this condition is sufficient, but in the general case, a suitable modification of this constraint is required. As an example it is enough to consider the system Δh = 0, h| ∂Ω = 1 and h → 0 as |x| → ∞, with Ω = R 3 \B 1 (0). Hence h = 1 |x| ; taking e = ∇h system (2.1) is fulfilled (with f ≡ 0), but of course ∂Ω e · n dσ = 0 does not hold.
Hence we reduced our consideration to analysis of the well posedness of system (2.1).
To analyze precisely regularity of solutions in a domain Ω we are required to construct a suitable localization. Given λ > 0, a parameter which will be specified later (and may depend on the point at the boundary), we choose a compact set K such that Then we introduce a smooth function η 0 : The second step are considerations near the boundary. Given x 0 ∈ ∂Ω we find η x0 : Vol. 16 (2014) The Rot-Div System 705 We see that Moreover, for fixed λ > 0 there exists a finite subcovering, i.e.
Next we introduce the basic function spaces [3,4,29]. By L p (Ω) we denote the usual Lebesgue space of functions integrable with the p-th power over the domain Ω. The Sobolev space W k p (Ω) is endowed with the following norm p,q , then ∇f ∈ B s−1 p,q and for positive index s one can define the space by induction as follows the modulus of continuity is defined by Just to point the connection to the standard modulus of continuity we note that Let us note that in general B 1 p,q (R n ) = W 1 p (R n ), more precisely, for p = 2 we can not find q to fulfill the identity. Moreover, we used Δ 2 h , instead of single Δ h , to cover the limit case s = 1. Next, we recall that for any domain Ω with sufficiently smooth boundary (Lipschitz continuity is enough), the Besov space B s p,q (Ω) stands for the restriction (in the distributional meaning) of functions in B s p,q (R n ) to Ω. That is f ∈ B s p,q (Ω) means that there exists somef ∈ B s p,q (R n ) such that for any smooth function ϕ with compact support in Ω we have where the infimum is taken over all the functionsf such that (2.6) holds, it endows the set B s p,q (Ω) with a structure of Banach space. We refer here to [5,7,29].
We also recall that the Besov spaces are a real interpolation family, namely whenever 1 ≤ p, q, q 1 , q 2 ≤ ∞, s 1 = s 2 and θ ∈ (0, 1). A great deal of our results will be based on the following interpolation property [3,29]: Note that (2.8) defines the Besov spaces in indirect way, which, however, is often very practical.
The following density and duality results will be used several times (see [5,29]) , where 1/p = 1 − 1/p and 1/q = 1 − 1/q. We will also use that functions in B s p,q (Ω) with s > 1/p have a trace at the boundary, see e.g. [29]. In order to make the above statement more accurate, we explain what a Besov space on the boundary is. In fact, Besov spaces may be defined on any r-dimensional manifold S. The idea is to use diffeomorphic maps after localization in order to reduce the definition to that of Besov spaces on R r (see e.g. [4,25,29]). If s ∈ (0, 1) and p = q ∈ (1, ∞) then the Besov space B s p,p (S) (which will be alternatively denoted by W s p (S) in some places of the paper) may be endowed with the norm where · Ḃs p,p(semi) (S) stands for the following homogeneous seminorm: The above double integral may be also used to define the homogeneous Besov spacesḂ s p,p (Ω) anḋ B s p,p (∂Ω). We shall be careful with this definition, since it has different properties for different types of sets S. The main examples are S = R n and S = Ω, where Ω is a subset of R n with boundaries (bounded, or the halfspace).
More precisely, for 0 < s < 1 p and 1 < p < ∞, we define the spaceḂ s p,p (Ω) aṡ Vol. 16 (2014) The Rot-Div System 707 This point requires some precision. The quantity (2.10) is indeed not a norm, except for the case S = R n . This follows from the fact (2.10) does not see constants. Thus the best solution is to consider the extension of a function onto the whole space; we call it Ef for a function f defined on Ω. Then we put with 1 p − 1 q = s n (for s ∈ (0, 1 p ) the number q is finite). The r.h.s. is the inhomogeneous norm, hence it allows to extend the function on the whole space, and this fact explains relation (2.12).
For such small s the above space is a Banach space, additionally we obtain the spacesḂ s p,p (Ω) for −1 + 1 p < s < 0 by duality: we setḂ s p,p (Ω) := (Ḃ −s p ,p (Ω)) * . The remaining spaces B s p,q (Ω) for 1 < p < ∞, 1 ≤ q ≤ ∞ and −1 + 1/p < s < 1/p may be defined by interpolation according to the following relation: We just have to fix some −1 + 1/p < s 1 < s 2 < 1/p and take θ ∈ (0, 1) such that s = θs 2 + (1 − θ)s 1 . To have the full picture of possible definitions of Besov space we refer to Triebel's book [29]. We now introduce some auxiliary results. First let us show the following Proof. Note that due to the assumption n · f = 0 at the boundary it is possible to extend f on the whole space by zero, preserving its divergence free property. Let us denote the extension by F . Next, we take R sufficiently large and examine the function F χ BR . Indeed, in general, div F χ BR = 0, thus we correct this truncation. The trace n · F | ∂BR(0) may not be zero, hence we ought to solve the following problem Since div F = 0 in R 3 , and ∂B R (0) may be treated as a boundary of sets B R (0) or R 3 \B R (0), we find As F ∈ L p (R 3 ; R 3 ), so the r.h.s. of (2.16) goes to zero as R → ∞. Solution to system (2.15) gives that So we take Then div F R = 0 in R 3 , since by definition n · F R = 0 at ∂B R (0). Additionally, by (2.17) we find In general F R may not be smooth, so we mollify it introducing with ω the standard mollifier (ω → δ in D (R 3 ) as → 0 + ). To finish, we put where R( ) → +∞ for → 0 + . Then for a suitable choice of R( ), function f fulfills (2.14). The a priori estimate of solutions to the second order elliptic problems will fundamentally depend on results for the Laplace operator in the halfspace. (

2.24)
Proof. In order to prove inequalities (2.23) and (2.24), let us observe that they follow from the results for scalar equations for the model problems in the half space. The boundary condition in that case can be split into three independent parts. Due to the basic properties of the Laplace operator we are able to solve model problem in the half space via symmetry. In the case of Dirichlet boundary conditions we admit skew-symmetric extension of the problem getting a system in the whole space, then the standard Marcinkiewicz multiplier theorem [19,29] yields that solution to moreover, the function u is locally integrable. Additionally, if h ∈Ḃ s p,q (R 3 + ) with −2 + 1 p < s < 1 p , we extend the function in the skew-symmetric way keeping the desired class of regularity. Namely, the skew-symmetric subspace ofḂ s p,q (R 3 ) for such s is a well defined subspace of the dual space toḂ −s p ,q (R 3 ). Thanks to the symmetry we gain one derivative more. See details in [7] and [21]. Then we get (

2.27)
A weaker result we get for the Neumann problem: (2.28) We use symmetric extension to R 3 for g = 0 and extension of g to R 3 + if g = 0. In the case of the simple L p spaces with the compatibility condition R 3 + h dx = R 2 g dx (if they are defined) we get and for h ∈Ḃ s p,q (R 3 + ) with −1 + 1 p < s < 1 p (the condition on s is more restrictive than for the Dirichlet boundary conditions) The last two lemmas concern multiplication in Besov space in the case as one of the functions has a compact support.

31)
Vol. 16 (2014) The Rot-Div System 709 Proof. The proof of the above result will be a consequence of the interpolation theorem and two estimates proved for the Besov spacesḂ s p,p (Ω). We refer also to [25]. Let 1 < p < ∞ and 0 < s < 1 p , then fg Ḃs In order to prove (2.32) we apply the integral definition of the Besov spaces (2.12). Because of the support of g, we conclude that fg ∈Ḃ s p,p (B λ (0)). Hence first we consider theḂ s p,p(semi) norm. We have (2.33) We consider the first integral with 3 s ( 1 p − 1 p * ) = 1; note that p * is finite, because s < 1 p . The second integral from (2.33) is estimated as follows This finishes the proof of (2.32).
To show (2.37) we just reduce the proof to an application of (2.32) and duality argument. By the definition of spaces with negative power, we find where the supremum is taken over all h ∈Ḃ s q,q (R 3 ) with the norm less or equal 1. As s < 1 − 1 p = 1 q , we may consider only functions supported in the ball. Taking into account results (2.32) and (2.37), using (2.13) we find (2.31). Lemma 2.7 is proved. (2.39) Proof. As usual in this limit case we will apply indirect technique of interpolations. Let us assume that in the below paragraph g ∈ C ∞ 0 (B λ (0)); hence it may be extended by zero to the whole R 3 . We show that if g ∈Ḃ 1+s p,p (R 3 ), then g ∈Ḃ s p,p (R 3 ) and Note that we easily find the following bounds: for s ∈ (0, 1). Here spacesẆ t p andḂ t p,p are closures of smooth functions with support bounded in B λ (0). From (2.41) and (2.42) we find (2.40). Subsequently we consider the case with negative powers. Here we will assume that s is near zero. Let g ∈Ḃ 1−s p,p (R 3 ); we want to show that g ∈Ḃ −s p,p (R 3 ). We have by the dual definition of the norm where the supremum is taken over all h ∈Ḃ s q,q (B λ (0)) ∩ { B λ (0) hdx = 0} with the norm less than one. Here we remind that the support of g is in B λ (0) and the taken assumption gives B λ (0) g dx = 0. Next, let us observe that considering two imbeddings the interpolation gives us On the other hand, h ∈ L q * (B λ (0)) with 3 s ( 1 q − 1 q * ) = 1. As s < 3 q = 3(1 − 1 p ), then q * is finite. Hence we get Taking (2.40) and (2.47) together, the interpolation theorem (see (2.13)) yields

Basic Existence in the L 2 -Framework
The first part which needs some explanation is the existence result for (a smooth compactly supported) right-hand side in the Hilbert space setting. We want to use this result in order to establish existence of solutions (for smooth and compactly supported r.h.s). This point is very important, since it enables to construct a sequence of approximative solutions for general right-hand sides. We aim at proving the following result: Vol. 16 (2014) The Rot-Div System 711 Theorem 3.1. Let Ω be as in Theorem 1.4. Let f ∈ L 2 (Ω; R 3 ), div f = 0 in D (Ω), in addition let f have a compact support. Then there exists a unique solution to (2.1) such that e ∈ W 1 2(loc) (Ω; R 3 ) and where C depends on the measure of support of f . Section 3 is devoted to the proof of this theorem. To fix the functional setting, we introduce , div u = 0 in Ω and u · τ k = 0, k = 1, 2, at ∂Ω .
We first start with the case ∂Ω ∈ C 2 , to avoid certain technical steps. We return to the case of less regular boundary at the end of this section. We have Lemma 3.2. Let Ω be as above with ∂Ω ∈ C 2 . Then there exists C > 0, depending on the measure of the compact subset K ⊂ Ω, supp e ⊂ K, such that for any e ∈ H the following inequality holds Proof. The first point which should be clarified is the information coming from the boundary condition.
Due to the smoothness of e up to the boundary we have div e = 0 also on ∂Ω. Taking a fixed point Here n(x 0 ), τ 1 (x 0 ), τ 2 (x 0 ) are fixed constant vectors defined by the frame at tangent space at point x 0 . From e · τ k = 0 we find (e · τ k ) τ k = 0, so e τ k · τ k + e · τ k, τ k = 0 at ∂Ω; (3.5) looking at the point So we have ∂(e · n) ∂ n = e · n(χ 1 (x 0 ) + χ 2 (x 0 )). (3.7) Recalling the well known identity rot rot e = ∇ div e − Δe, multiplying it by e, integrating over Ω and performing integration by parts we get Hence by (3.7) and the condition for the tangent value at the boundary we find Here we assume that χ k is bounded (∂Ω ∈ C 2 ), but we return to this issue later. The above analysis yields the following relation By (3.11) and (3.13) we are able to ensure that e k | K L2(K) = 1; (3.14) so By (3.14) and (3.15) there exists a function e * such that up to a subsequence ∇e k ∇e * weakly in L 2 (Ω; R 3×3 ). (3.16) But (3.13) yields that rot e * ≡ 0, moreover ∂Ω e k · n dσ = 0, hence the limit e * fulfills (in the weak sense, using the well-known operator identity Δ = ∇ div − rot rot and the boundary conditions) The only solution to (3.17) is e * ≡ 0, see Remark 1.2. On the other hand by (3.14) and (3.15) (up to a subsequence) e k | K → e * | K strongly in L 2 (K), which implies that e * | K L2(K) = 1, leading to a contradiction. Hence (3.12) is true.
Due to this lemma we may define . (3.18) We easily show that Let us note that the condition ∂Ω u · n dσ = 0 is for u ∈ H fulfilled automatically since the spaceH consists only of functions with compact support, it is they are zero away a sufficiently large ball. Then the divergence free constraint implies this restriction. Therefore By the definition of H the functional b is elliptic, so we shall check whether a functional generated by f is bounded. But f is compactly supported, so where in the last step we used (3.12). The lemma is proved.
Vol. 16 (2014) The Rot-Div System 713 We want to extend the meaning of the solution, make it a distributional one by replacing H by smooth functions with just boundary condition for the tangent part of functions (zero).

Low Regularity Boundary Case
Let us now consider the case of less regular boundary, i.e. ∂Ω ∈ B only. In particular the curvature is a distribution. On the other hand, e ∈ H 1 (K; R 3 ), so to repeat all steps of the proof of Lemma 3.2 it remains to show The regularity of g implies that for a given > 0 we are able to find a smooth function g such that Finally, as g is smooth, → C 1,a for some a > 0, hence we get (3.30) also in this case, the proof is even somewhat easier than in the case considered above for the critical regularity space.

The L p Case
We start with the proof of the global estimate for (2.1) in the standard L p framework. The central point in our considerations is the regularity of the boundary. The structure of our estimates requires that with q fulfilling the conditions: if p > 3 then q = p and if p ≤ 3 then q > 3 (can be arbitrarily close to 3). Recall that in this case, due to the embedding, we know that the local description of the boundary is of the class C 1,a for some a positive. Recall also that W Proof. We show (4.2) by localization. Let η 0 be a cut off function "removing" the neighborhood of the boundary of Ω and supp ∇η 0 ⊂ K, see Sect. 2. Then we get in the sense of distributions with E = η 0 e. Therefore, immediately Note that even though this estimate was shown just formally, it can be easily justified also for our solution which was a priori only in W 1 2(loc) ; indeed, we get our estimate for any 1 < p ≤ 2. Hence, taking p = 2 we know that e belongs to W 1 6(loc) (Ω; R 3 ) and we may repeat the argument above for any 2 < p ≤ 6; hence taking p = 6 we know that e ∈ W 1 ∞(loc) (Ω; R 3 ) which allows us to consider also 6 < p < ∞.
Vol. 16 (2014) The Rot-Div System 715 Subsequently we consider the case near the boundary. Let η 0 , η 1 , . . . , η N be the partition of unity in Ω, see Sect. 2. Take now k ∈ {1, . . . , N}. As before, we start with a formal argument and then explain its application for e ∈ W 1 2(loc) (Ω) only. We have for E = η k e ΔE = 2∇η k · ∇e + eΔη k + η k f = f 1 in Ω, E · τ l = 0 on ∂Ω, l= 1, 2, div E = e · ∇η k on ∂Ω. (4.5) The above system is supported on the set supp η k ; we transform it into R 3 + getting (cf. Sect. 2) Δ z Z * E = (Δ z − Δ x )Z * E + Z * f 1 in R 3 + , Z * E ·ê l = 0 on R 2 , (Z * E) (3) ,z3 = Z * (e · ∇η k ) − (div x − div z )Z * E on R 2 , (4.6) where Z * E = E • Z andê l , l = 1, 2 be the unit vectors in the direction of the z l axis. Therefore (see Lemma 2.6) ). (4.7) However, let us observe that In order to prove the above estimate we note that (4.9) To avoid unnecessary notation we write in a symbolic way that ∇ x = (Id + A)∇ z , where A(0) = 0 and A ∈ W 1 q (B λ (x 0 ); R 3×3 ) (4.10) with q = p for p > 3 and q > 3 for p ≤ n. Note that A is Hölder continuous due to the embedding theorem. Using the above notation we get for some a > 0. To estimate the last norm we see that for p > 3 we have . In the case p ≤ 3 note that 1 p = 3−p 3p + 1 3 . So if q > 3, there is a room to apply the interpolation inequality to get In a similar way we examine the below estimate, note that the considered norm can be viewed as a trace norm, thus the computations for (4.10) stay true and we conclude Finally Z * (e · ∇η k )| z3=0 (4.14) Hence we obtain estimate (4.2).
Remark 4.2. The computations above were only formal as we had a priori no information about the existence of second derivatives of e or E. However, recalling the uniqueness we may replace (4.6) by the following problem Δ z Z * E = (Δ z − Δ x )Z * E + Z * f 1 in R 3 + , Z * E ·ê l = 0 on R 2 , (Z * E) (3) ,z3 = Z * (e · ∇η k ) − (div x − div z )Z * E on R 2 , (4.15)