Classical solutions of the divergence equation with Dini-continuous datum

We consider the boundary value problem associated to the divergence operator with vanishing Dirichlet boundary conditions and we prove the existence of classical solutions under slight assumptions on the regularity of the datum.


Introduction
In this paper we deal with the existence of classical solutions for the boundary value problem div u = F in Ω, namely we look for solutions u : Ω → Ê n , such that u ∈ C 1 (Ω) ∩ C 0 (Ω).
Here Ω ⊂ Ê n is a smooth and bounded domain, while F is a given continuous function satisfying the compatibility condition Ω F (x) dx = 0. This is a classical problem in mathematical fluid mechanics, strictly connected with the Helmholtz decomposition and the div-curl lemma (see Kozono and Yanagisawa [19]). We recall that by dropping the boundary condition a solution of the first order system (1) can be readily obtained by taking the gradient of the Newtonian potential of F . These aspects are extensively covered in Galdi [14,Ch. 3] with special attention to the work of Bogovskiȋ [7], where the problem is solved in the Sobolev spaces H 1,p 0 (Ω). Further developments may also be found in Borchers and Sohr [8]. For different approaches and results the reader could consider the books by Ladyžhenskaya [20] and Tartar [24], especially regarding the solution in the Hilbert case, while Amrouche and Girault [1] used the negative norm theory developed by Nečas [21].
Our approach follows closely the Bogovskiȋ's one, where the representation formula (2) below, in analogy with the "cubature" formulae of Sobolev, gives explicitly a special solution of the problem (1), which per se has infinitely many solutions. The formula (2) turns out to be extremely flexible in the applications to many different settings as, for instance, the recent results for weighted L p(x) -spaces (see Huber [16]). Classical results in Hölder spaces have been shown in Kapitanskȋ and Piletskas [17], as a corollary of a more general result, which seems to be obtained in a way different from ours. For the Hölder case see also the recent review in Csató, Dacorogna, and Kneuss [11]. In addition, we also note that the non-uniqueness feature of the first order system (1) allows some existence results with more regularity than expected from the usual theorems, as the striking results of Bourgain and Brezis [9], coming from a non-linear selection principle of the solutions to a linear problem (see the extensions to the Dirichlet problem in Bousquet, Mironescu, and Russ [10]).
Our interest for the problem is twofold: on one side, we want to investigate the results close to the limiting case F ∈ L ∞ (Ω) ∩ C(Ω), for which counterexamples to the existence of a solution are known (see Bourgain and Brezis [9] and Dacorogna, Fusco, and Tartar [12]); on the other side, we are interested in relaxing as much as possible the assumptions needed to get classical solutions. This is motivated by the aim of finding weaker sufficient assumptions which allow to construct classical solutions to fluid mechanics problems. Since the continuity of F is not enough to that purpose, we went back to the very old results by Dini [13] and Petrini [22] about the Poisson equation, and we consider the problem with the additional hypothesis that F is Dini-continuous (see below for a definition). Our proof follows closely an argument used by Korn to obtain a similar regularity result for the second derivative of the Newtonian potential (see Gilbarg and Trudinger [15,Ch. 4]) and exploits the property of the Dini continuity to "regularize" the singularity in the second derivative of the potential, outside the theory of Calderòn-Zygmund of singular integral operators. About the boundary condition, our proof is based on some "new" insight of the formula (in the sense that we made some simple observations on the Bogovskiȋ formula which we cannot find stated explicitly elsewhere in literature) still valid when the datum F cannot be approximated with compactly supported smooth data. Such an approximation seems to play a fundamental role in Sobolev or Orlicz spaces.
We wish to mention that the link between Dini continuity and existence of classical solutions in fluid mechanics started with Shapiro [23] in the steady case and found a very interesting application with the paper of Beirão da Veiga [2], where the 2D Euler equations for incompressible fluids are solved in a critical spaces for the vorticity. More recently, the same results have been also employed by Koch [18] and in [5] to study fine properties of the long-time behavior of the Euler equations. Finally, the interest for classical solutions of the Stokes system has been revived in the recent papers of Beirão da Veiga [3,4], that provided a further motivation to our analysis of the divergence and curl operator, since they are among the building blocks of the theory. We also point out that the system (1) is not elliptic, hence the well-known results for elliptic equations and systems do not apply directly.
To conclude we also mention that the problem of classical solutions for the curl equation (again with Dirichlet condition) will be treated elsewhere [6] following the same method, by using similar (but more complicated) representation formula, known in that case.

Notation and preliminary results
In this section we recall the main definitions we will use, as well as some basic facts about the representation formula developed by Bogovskiȋ. The results of this section are well-known, but some of them on the role of the boundary condition are not explicitly available in the literature.
In the following we denote by B(x, R) = {y ∈ Ê n : |y − x| < R}, the ball of radius R centered at x, by S n−1 = {y ∈ Ê n : |y| = 1} the unit sphere of Ê n , and by |S n−1 | its (n − 1)-dimensional measure.
We also denote by C D (Ω) the space of the (uniformly) Dini-continuous functions F, i.e., such that if one introduces the modulus of (uniform) continuity ω(F, ρ) = sup the function ω(F, ρ)/ρ is integrable around 0 + . We equip the space of Dini continuous functions with the following norm and it turns out to be a Banach space. We remark that, by the uniform continuity, any function in C D (Ω) may be extended up to the boundary of Ω with the same modulus of continuity. We observe also that C 0,α (Ω) ⊂ C D (Ω) for all 0 < α ≤ 1 and recall that its relevance in partial differential equations comes from the result that, if

Bogovskiȋ's formula and its variants
The aim of this section is to provide a representation formula for a solution of the divergence problem, due to Bogovskiȋ [7], as well as several useful variants and consequences. Unless differently specified (namely, in the last section), the following hypotheses will be tacitly assumed throughout all the paper. Let B denote the open unit ball of Ê n , n ≥ 2, centered at the origin. The scalar function ψ ∈ C ∞ 0 (Ê n ) with supp ψ ⊆ B, and it is not vanishing identically. We will denote by ∂ j ψ the partial derivative of ψ with respect to its j th argument.
The domain Ω will be a bounded open subset of Ê n , star-shaped with respect to any point of B.
The main results to be proved are the following Theorems.
Theorem 1. Assume the previous hypotheses. Let q > n and let F ∈ L q (Ω). Then: defines for any x ∈ Ê n (and not only almost everywhere) a function v : Ê n → Ê n ; ii) v(x) = 0 ∀ x ∈ Ê n \Ω; iii) For any q > n where c depends only on n, ψ, diam Ω, and q; Important consequences of the Theorem 1 are the next regularity results for the "potential" v in the interior, as well as at the boundary.
The proofs require several lemmas, and we start with an estimate for the kernel that appears in the Bogovskiȋ's formula (2).
Definition 4. Let ψ be as above. Then we set and we remark that we can rewrite the Bogovskiȋ's formula as follows There exists a constant c > 0, depending only on n, ψ, and diam Ω, such that Proof. Since the smooth function ψ vanishes outside B, then it follows that ψ y + ξ x−y |x−y| = 0, when y + ξ x−y |x−y| ≥ 1. Since y + ξ x−y |x−y| ≥ |ξ − |y| |, we have that ψ y + ξ x−y |x−y| is zero for ξ ∈ Ê + such that ξ > 1 + diam Ω, and therefore +∞ |x−y| and the lemma follows.
The following simple remarks have important consequences in the study of the support of v.
Proof. In fact, since ψ is not zero, if follows that Moreover, x = y + |x − y| x−y |x−y| and hence x belongs to the segment of endpoints y and y + ξ x−y |x−y| , for any ξ > |x − y|. If it were y ∈ Ω, by the hypotheses on Ω the entire segment would lay in it, and that contradicts the assumption on x.
An immediate consequence of the last lemma is the following result ∈ Ω, and ∀ y ∈ Ω We can now give the proof of Theorem 1 Proof of Theorem 1. By the above lemma, the vector v(x) vanishes outside Ω, and then ii) follows from i).
Since Ω is bounded, by Lemma 5 it turns out that N (x, ·) ∈ L p (Ω) and by Hölder's inequality it follows that and so i) and ii) follow.
Finally, by setting g(z) = |z| 1−n , we have where the right side is independent of x. Since, by ii), v vanishes outside Ω, it follows immediately iii).
To get iv), it is enough to put ξ = α|x − y| in the initial formula. By putting r = ξ − |x − y|, instead, it follows v). Finally, by introducing z = x − y in v) it follows vi).
Remark 8. It is useful to remark explicitly that the Bogovskiȋ's "potential" v vanishes at the boundary for any F ∈ L q (Ω), without any other assumption than those made on Ω and ψ in Theorem 1. It is relevant to observe that it does not come by approximating F by C ∞ 0 (Ω) functions and by taking limits, but it is a property which descends directly from the formula for a large class of functions. In our case that is especially useful in that a given function in C D (Ω) cannot be approximated in uniform norm by regular function with compact support, unless it vanishes at the boundary.
Proof of Theorem 2. From the formula in Theorem 1 vi), it follows that Since ψ and F , as well as all their derivatives of any order, are bounded on Ê n , the integrand is bounded by a multiple of the function |z| 1−n , which is integrable on Ω. By differentiating inside the integral, it follows v ∈ C ∞ (Ω). Finally, in order to get supp v ⊂ Ω, let which is a compact subset of Ω by the hypotheses, and fix any x ∈ Ω\E. Now, it will be shown that y + r(x − y) / ∈ B for any y ∈ supp F and any r > 1. In fact, as it has been already seen in the proof of Lemma 6, x belongs to the segment of endpoints y and y + r(x − y) for any r ≥ 1. Thus, if y + r(x − y) ∈ B, it would follow x ∈ E, that contradicts the assumption on x.
for some q > n, set to zero outside Ω, and let v k and v the corresponding values obtained by the Bogovskiȋ's formula. By Theorem 1 ii) and iii), it follows that v k converge uniformly to v in Ê n . Since, by Theorem 2, , the theorem follows immediately.

Further properties of the kernel
In this section we prove some properties of the kernel appearing in the Bogovskiȋ's formulas which turn to be useful in the following. Next lemma provides an identity about the derivatives of the kernel as it appears in the second formula (Theorem 1 iv)).
Lemma 9. For any fixed x, y ∈ Ω, x = y let Then, it follows that Proof. Since |y + α(x − y)| ≥ 1 for α ≥ (1 + |y|)/|x − y| the integrand is bounded on a compact subset of Ê. By differentiating under the sign of integral, it follows that and hence the lemma.
The following result provides the fundamental estimates on ∂ x j N i (x, y), which allow to exploit the Calderòn-Zygmund theory to obtain the original Bogovskiȋ's results about the H 1,p 0 (Ω) regularity of v, and the Dini continuity hypothesis to prove the results below.
Theorem 10. For any i, j = 1, . . . , n there exist functions K ij and G ij such that where K ij (x, ·) is a Calderòn-Zygmund singular kernel and G ij is a weakly singular kernel in the sense that, if one sets then, there exist constants c = c(ψ, n) and M = M (ψ, n, diam Ω) such that: Proof. Fix x, y ∈ Ω, x = y. By introducing r = α|x − y| − |x − y| it follows that By using the binomial expansion inside the integrals, the last sum may be written as follows involves only the terms not containing any strictly positive power of |x − y|, while all of the others, grouped as G, contain at least a factor |x − y| coming from the expansion; therefore, G ij verifies the estimate in iv).
The homogeneity property in i) follows immediately from the previous expression of K ij .
Furthermore, z ∈ S n−1 implies and ii) follows. Finally, After an integration by parts, the last integral turns out to be zero, and iii) is proved. Finally, ii) and iv) imply immediately v), on the bounded set Ω.

The approximating functions for the solution
The main tool we applied in this paper is an old aged argument exploited by Korn (see, e.g., Gilbarg and Trudinger [15,Ch. 4]) in the study of the existence of classical solutions of the Poisson equation, based on a suitable "cutoff" of the singularity present in the second derivatives of the Newtonian potential, that provides a way to approximate the solution v by regular functions.
To this end we introduce the function η ∈ C ∞ (Ê + ) such that η(t) ≡ 0 Definition 11. For any q > n, F ∈ L q (Ω) and ǫ > 0 let us set We remark that if |x − y| < ǫ the integrand is zero and therefore the integrand belongs to C ∞ (Ê n ) and it is bounded by Lemma 5, while if |x − y| ≥ ǫ the set of α such that |y + α(x − y)| ≤ 1, where ψ could be not null, is bounded as well. It follows immediately that v ǫ is well-defined and belongs to C ∞ (Ê n ). Moreover, under the same hypotheses of Theorem 2 and following the same argument, one proves that it belongs actually to C ∞ 0 (Ω). Theorem 12. Assume the same hypotheses of Theorem 1, and let η and v ǫ as above. Then: ii) If, moreover, Ê n ψ(x) dx = 1 and F ∈ C 0 (Ω), then Proof. To prove i), let us fix any x ∈ Ω. Remark that, by the Lemma 5, for any ǫ < dist(x, ∂Ω) and by the absolute continuity of the last integral, it follows than it tends to zero independently of x ∈ Ω. To complete the proof of i) it is enough to remark that by Lemma 7 v To prove ii), by differentiating v ǫ i at any x ∈ Ω it follows that

Now,
as ǫ tends to zero. Moreover, by setting into C first α = ξ/|x − y|, next ξ = r + |x − y|, and finally z = ǫ −1 (x − y) one obtains Now we claim that, as ǫ goes to 0, the last term tends to In fact, since ψ x + r z |z| vanishes when r > 1 + diam Ω, the inner integral is bounded by max |ψ| (1 + diam Ω) n =: M . Hence, we get where ω is the modulus of continuity of F . By the uniform continuity of F on Ω and the Lebesgue theorem on dominated convergence, the last integral vanishes as ǫ goes to zero and therefore the claim is proved. Finally, by introducing in the limit (⋆) the radial and angular coordinates ρ = |z| and u = z/|z|, one gets Therefore, C −→ F (x) as ǫ tends to 0 and the lemma follows.
The following theorem, an immediate corollary of the previous one, is the cornerstone of the resolution of the divergence problem.
Theorem 13. Assume the same hypotheses of the previous theorem. Moreover, let Ê n ψ(y) dy = 1, F ∈ C 0 (Ω) and Ω F (x) dx = 0. Then, as ǫ goes The next lemma, an immediate consequence of Lemma 9 and the opposite sign in the derivatives of η(|x − y|/ǫ), will be useful in proving the subsequent representation formula.
As usual in potential theory, getting a representation formula for the derivatives of the function v i is a crucial goal. We will obtain it through a limit of the derivatives of its "regular approximation" v ǫ . Thus, let us start by differentiating the formula where F ∈ L ∞ (Ω) is extended by zero outside Ω and B R is a ball of radius large enough such that Ω ⊂⊂ B R . By the previous lemma and Bogovskiȋ's formula in Theorem 1 iv) it follows that Since η |x−y| ǫ = 1 if ǫ < dist(∂B R , Ω), by the Gauss-Green formula the last integral is equal to ∂B R N i (x, y)ν j (y)dσ y .
The previous computation suggests to put forward a conjecture about the limit as ǫ goes to zero, which will be proved in the next theorem, that is the main result of the paper.
Theorem 15. Assume all the hypotheses of the Theorem 1, and let η and v ǫ as above. Furthermore, let ψ be such that Ê n ψ = 1, F ∈ C D (Ω), and Then: Proof. To prove i), fix any x ∈ Ω. Remark that, after its extension by zero outside Ω, F ∈ L ∞ (Ê n ). For any ǫ < dist(x, ∂ Ω) one has Since B(x, ǫ) ⊂ Ω, by Theorem 10 v) it follows that where ω is the modulus of continuity of F in Ω. By introducing the radial and angular coordinates, the last integral becomes and, by the Dini continuity hypothesis on F , it is finite. Furthermore, since both F and ∂ x j N i (x, y) are bounded on {|x− y| ≥ ǫ}, the term C is finite as well.
Finally, since ∂ j ψ ∈ C ∞ 0 (Ê n ) and supp ∂ j ψ ⊂ B, it follows that is the value of Bogovskiȋ's formula corresponding to the bounded function F ≡ 1, evaluated by using ∂ j ψ instead of ψ. By Theorem 1 iii), it is globally bounded, and i) follows. To prove ii), fix any Ω ′ ⊂⊂ Ω. Thus, for any x ∈ Ω ′ and ǫ > 0 such that 2ǫ < dist(Ω ′ , ∂Ω), it follows that As above, by Theorem 10 v) it follows that By the Dini continuity of F and the consequent absolute continuity of the integral, the last term vanishes as ǫ goes to zero, independently of x ∈ Ω ′ . In order to estimate the second term E remark that, by Theorem 1 v) and the hypothesis on By introducing the variable y = x + ρu, since it follows as above that the last term is bounded by a multiple of 2ǫ ǫ ω(F,ρ) ρ dρ and, again by the absolute continuity of the integral, E vanishes as ǫ goes to zero, independently of x ∈ Ω ′ . Finally, by using ∂ j ψ instead of ψ as in the proof of the previous i), from Theorem 1 iii) it follows that for any q > n and suitable constants c ′ , c ′′ .
Since the last norm vanishes as ǫ goes to zero, for any q > n and independently of x ∈ Ω, ii) follows.
From ii), by the classical theorem on a convergent sequence of functions whose derivatives converge uniformly, it follows iii), while iv) follows immediately from ii), iii), since v ǫ ∈ C ∞ (Ê n ).
All the previous results lead to the following theorem. Then, for any F ∈ C D (Ω) verifying Ω F = 0, the Bogovskiȋ's formula Proof. At first observe that, if B = B(0, 1), the theorem follows immediately from Theorem 1 ii), Theorem 3, Theorem 13 and Theorem 15.
Otherwise, let us set z = (x − x 0 )/R, Ω = {(x − x 0 )/R : x ∈ Ω}, F (z) = F (x 0 + Rz) and remark that Ω and F fulfil the previous hypotheses with respect to B = B(0, 1). Now, let w(z) be the solution of div w(z) = F (z) in Ω, w ≡ 0 on ∁ Ω, whose existence follows by the initial observation, and remark that is the requested solution, and the proof is completed.
3 Existence of classical solutions for the divergence problem in more general domains.
The aim of this brief section is to relax the very strong geometric restriction on the domain Ω requested in the above result, although at the price to renounce the simplicity of a single Bogovskiȋ's representation formula for the solution of the divergence problem. The next theorem provides the existence of a classical solution in a wider class of domains including, for instance, those with a smooth boundary. To this aim, we start to prove a suitable "partition of unity" lemma. of Ω such that, if one sets Ω i := Ω ∩ G i it follows that: • Ω i is star-shaped with respect to every point of an open ball B i , with B ⊂ Ω for any i = 1, . . . , m + p; • ∂Ω ⊂ ∪ m 1 G i ; • G i is an open ball with closure in Ω for any i = m + 1, . . . , m + p; • Ω = ∪ m+p Proof. The proof if this result may be obtained as in Galdi [14,Lemma III.3.4], by replacing C ∞ 0 with C D in any occurrence involving f , f i or g i , by assuming Ω as their domain, and by extending ψ i and χ i by zero outside their supports.
Remark 18. We remark explicitly that from i) and iii) it follows immediately the crucial property which, together with the properties of the covering, allows to apply the regularity result in Theorem 16 to the divergence problem "localized" at Ω k .
The final result, which extends Theorem 16 to a considerably wider class of domains, will be now obtained by a localization argument.

Theorem 19.
Let Ω be a bounded subset of Ê n with a locally Lipschitz boundary. Then, for any F ∈ C D (Ω) with Ω F (x) dx = 0 there exists a solution v ∈ C 1 (Ω) ∩ C 0 (Ω) of the problem div v(x) = F (x) in Ω, v ≡ 0 on ∂Ω. Proof.
Let Ω k and F k be defined as in the previous lemma, and let v k be the solution in C 1 (Ω k ) ∩ C 0 (Ê n ) of the problem and since by Lemma 17 i) and ii) the last term is equal to F (x), and then v is the aimed solution.