On the regularity of very weak solutions for linear elliptic equations in divergence form

In this paper we consider a linear elliptic equation in divergence form 0.1∑i,jDj(aij(x)Diu)=0inΩ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$\end{document}Assuming the coefficients aij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ij}$$\end{document} in W1,n(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,n}(\Omega )$$\end{document} with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution u∈Llocn′(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in L^{n'}_\mathrm{loc}(\Omega )$$\end{document} of (0.1) is actually a weak solution in Wloc1,2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,2}_\mathrm{loc}(\Omega )$$\end{document}.


Introduction
Let n ≥ 2 and Ω ⊂ R n be a bounded open set. In this paper we study regularity properties of very weak solutions to the linear elliptic equation where the matrix-field A : Ω → R n×n , A(x) = (a ij (x)) i,j , is elliptic and belongs to W 1,n (Ω, R n×n ) ∩ L ∞ (Ω, R n×n ), i.e. Finally we assume that the coefficients (a ij (x)) i,j are double-Dini continuous in Ω, i.e. a ij ∈ C 0 (Ω) and

) is
A ∈ W 1,p (Ω, R n×n ), with p > n. On the other hand, condition (1.4) occurs not only for ω(r) = r α , but more generally for ω(r) = log β 1 r , β < −2. Given a measurable matrix A(x) = (a ij (x)) i,j satisfying (1.3), a function u ∈ W 1,2 loc (Ω) is called a weak solution of (1.1) if The celebrated result by De Giorgi in [5] states that if u is a weak solution of (1.1) then u is locally Hölder continuous.
Subsequently, J. Serrin produced in [14] a famous example, constructing an equation of the form (1.1) which has a solution u ∈ W 1,p (Ω), with 1 < p < 2, and u / ∈ L ∞ loc (Ω). Serrin conjectured that if the coefficients a ij are locally Hölder continuous, then any solution (in the sense of distributions) u ∈ W 1,1 loc (Ω) of (1.1) must be a (usual) weak solution, i.e. u ∈ W 1,2 loc (Ω). Serrin's conjecture was established by Hager and Ross [9], and then in full generality by Brezis [2] (see also [1] for a full proof) starting with u ∈ W 1,1 loc (Ω), or even with u ∈ BV loc (Ω), i.e., u ∈ L 1 loc (Ω) and its derivatives (in the sense of distributions) being Radon measures. Let us remark that in Brezis's result the coefficients a ij , satisfying (1.3), are Dini continuous functions in Ω. The Dini continuity of the coefficients is optimal in some sense: for the unit ball B 1 and continuous coefficients, Jin et al. [10] constructed a solution (in the sense of distributions) u ∈ W 1,1 loc (B 1 )\W 1,p loc (B 1 ) for every p > 1. For A(x) = (a ij (x)) i,j satisfying (1.2) and (1.3), we will consider a very weak solution u ∈ L n loc (Ω) of (1.1), namely with n = n n−1 . Remark 1.1. It is not difficult to prove that the test functions ϕ in (1.5) can be taken in W 2,n (Ω) ∩ W 1,∞ (Ω), with supp ϕ Ω. Indeed, one can argue by density to show that given a function ϕ ∈ W 2,n (Ω) ∩ W 1,∞ (Ω) with compact support, we may find a sequence ϕ k ∈ C ∞ c (Ω) such that ϕ k → ϕ strongly in W 2,n and sup k ϕ k 1,∞ < ∞ (so that Dϕ k converges to Dϕ weakly* in L ∞ ) and then taking the limit as k goes to infinite in the Eq. (1.5) for ϕ k . The main result of the paper is the following.  3), one can consider a very weak solution u ∈ L n loc (Ω) to (1.1), but when dealing with the regularity properties of u some extra conditions on the coefficients a ij must be considered. The counterexample constructed in [10] provides in fact continuous coefficients a ij which belong also to W 1,n (B 1 ), showing that one can not expect a very weak solution u ∈ L n loc (Ω) to be a weak solution in W 1,2 loc (Ω) under just conditions (1.2) and (1.3). For the sake of completeness, we will propose the example given in [10] in the "Appendix B", underlining that the constructed coefficients belong also to W 1,n (B 1 ).
On the other hand, in Sect. 4 we propose an alternative to double Dini continuous coefficients which again bypasses the counterexample. In particular, under hypotheses (1.2) and (1.3) we consider a very weak solution in L q loc (Ω), with q > n . Remark 1.4. In [15] Zhang and Bao deal with the case of very weak solutions u ∈ L 1 loc (Ω) of (1.5), interpreting the coefficients as Lipschitz functions, due to the assumption made on the solutions. Thus our result represents a natural extension from their research.

Notation and preliminary results
We collect here the main definitions and notation and some useful results that will be needed in the sequel.

Notation
In the following, we denote by B r (x) = {y ∈ R n : |y − x| < r} the ball of radius r centered at x.
We indicate by {e 1 , . . . e n } the canonical basis of R n . Given h ∈ R\ {0}, for a measurable function ψ : R n → R and for = 1, . . . , n, we introduce the notation for the incremental quotient in the -th direction. We recall that for every pair of functions ϕ, ψ, we have The following result pertaining to difference quotients of functions in Sobolev spaces is well known t (see [8,Proposition 4.8] for example).

Dini continuous functions
We say that a continuous function f on Ω is Dini continuous if the modulus We also denote by C D (Ω) the space of Dini continuous functions; it turns out to be a Banach space equipped with the following norm: where · ∞ is the usual uniform norm. Let us 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. Moreover, for any 0 < α ≤ 1, where C 0,α (Ω) denotes the space of Hölder continuous functions.
The space C D c (Ω) will denote the set of functions in C D (Ω) with compact support in Ω.
It is easily seen that f ε uniformly converges to f in Ω, thus in order to prove (2.2) we will just show that as ε tends to 0. Observe that which together yield which concludes the proof of (2.2).

Lemma 2.4.
Let f, f ε , and g belonging to C D (Ω) such that f ε converges to f in C D ; then gf ε converges to gf in C D .
Proof. As before, it is enough to prove the convergence of the seminorm since the uniform convergence is immediate. Then, writing the definition of the modulus of continuity, we have which goes to zero as ε tends to zero.

C 1 -Dini regularity of solutions to divergence form elliptic equations with
Dini-continuous coefficients For the proof of our result, we will need the following extension of the Schauder regularity theory for elliptic equations in divergence form with Dini continuous coefficients (see [12,Theorem 1.1] and [6, Theorem 1.3], see also [11] which is inclusive of the parabolic case.). For the L p -regularity theory we refer to [7], where the general case of V MO coefficients is treated (see also [13,Theorem 5.5.3 (a)] or [3, Theorem 2.2. Chapter 10] for the case of continuous coefficients).
Remark 2.6. The first conclusion of Theorem 2.5 comes with an estimate of the Dini modulus of continuity of Du involving the Dini modulus of continuity of a ij and f j . Actually, in [12, Theorem 1.1] and in [6,Theorem 1.3] only the continuity of Du is proved and these results are obtained with a weaker assumption on the coefficients a ij . Assuming (1.4) for the coefficients we are able to prove also the Dini continuity of the gradient of the solution. In "Appendix A" we will resume in broad terms the proof of [12, Theorem 1.1], developing it in order to get the needed Dini continuity result.

C 2 -regularity of solutions to non divergence form elliptic equations with
Dini-continuous coefficients Let us first recall the W 2,p -solvability of the Dirichlet problem for non divergence elliptic equations with discontinuous coefficients (see [4,  where Ω is a C 1,1 smooth and bounded subset of where the constant C depends on n, p, λ, Λ, ∂Ω, A W 1,n (Ω,R n×n ) .
The next result specifies estimate (2.7); its proof is quite standard but we prefer to write it for the sake of completeness.
with a ij , f, p and Ω as above. Then having in mind Theorem 2.7, if we prove that for any operator L ∈ L and for any f ∈ L p (Ω), the solution u of we are done. Suppose it is not the case, then this is equivalent to say that for every N ∈ N, there exists an operator L N = i,j a N ij D ij ∈ L and a function f N ∈ L p (Ω) such that the corresponding solution u N to the Dirichlet problem where C does not depend on N and hence, Thus v N is a precompact sequence: up to a non relabeled subsequence, we can suppose v N u * weakly in W 2,p (Ω), for some u * ∈ W 2,p (Ω), moreover u * ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω). Similarly, we can also say that, for every i, j = 1, . . . , n, a * ij weakly in W 1,n (Ω) and a N ij → a * ij strongly in L q (Ω) ∀ 1 ≤ q < ∞. Thus, the operator L * = i,j a * ij D ij belongs to L and for ϕ ∈ L p (Ω) we have ˆΩ Therefore, L N v N converges weakly in L p (Ω) to L * u * . On the other hand, using (2.9), we have Passing to the limit in the equation satisfied by v N , we discover that the limit u * ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω) satisfies L * u * = 0 a.e. in Ω. By the uniqueness properties of the solutions to (2.6), it follows that u * = 0. Thus v N converges to zero and the argument becomes contradictory since v N L p (Ω) = 1.
In [6,Theorem 1.5] it is shown that solutions to elliptic equations in non divergence form with zero Dirichlet boundary conditions are C 2 up to the boundary when the leading coefficients are Dini continuous functions. then u ∈ C 2 (Ω).

Remark 2.10.
The assumption in [6] about the coefficients is weaker then (1.4), since they assume that the modulus of continuitỹ On the regularity of very weak solutions Page 9 of 23 43

Proof of the main theorem
We use a duality argument in conjunction with the regularity properties for elliptic equations in divergence and in non divergence form, stated in Theorems 2.5 and 2.9.
Step 1 For = 1, . . . , n, we claim that Δ h u is bounded in the dual space of Dini continuous functions with compact support (C D c (Ω )) . Given a Dini continuous function w ∈ C D c (Ω ), according to Theorem 2.9 combined with Theorem 2.7, the solution v ∈ W 2,q (Ω 0 ), ∀q > 1, to the Dirichlet problem We fix one of these balls and the related function η k ; we omit to indicate the center x k and the index k for η k for simplicity.
In view of Remark 1.1, we can insert ϕ = ηΔ −h v in (1.5), getting With a simple change of variables, we get where we also used (2.1). Thus, we finally have The use of Hölder's inequality gives combined with Sobolev's embedding and Proposition 2.8 in the last inequality. Analogously The terms I 3 and I 4 can be treated in the same way. Using Hölder's inequality, Theorem 2.1 and Proposition 2.8, we have Again, for I 5 we have We finally estimate I 6 . From (2.1), we get The second term can be estimated as I 3 and I 4 , thus: Here we have used once more Theorem 2.9. Finally, combining the estimates found for I m , m ∈ {1, . . . 6}, from (3.2) we get for every w ∈ C D c (Ω ). By the uniform boundedness principle this means that {Δ h u} h is a family of equibounded elements in the dual space of Dini continuous functions (C D c (Ω )) . Since (C D c (Ω )) is separable, we have that, up to a subsequence, Step 2. We prove that u ∈ W 1,p loc (Ω), with p > n. Using the previous Step we can easily deduce from (1.5) that where the duality pairing is between (C D c (Ω )) and C D c (Ω ).
with p > n. Introducing as before a regular set Ω 0 between Ω and Ω we can possibly assume that Ω is a C 1,1 set. Let v ∈ W 1,2 0 (Ω) be the weak solution of the problem By Theorem 2.5 we have that v ∈ W 1,p 0 (Ω) and Note that, since p > n, this means also that the function v is Hölder continuous. We take B R/2 ⊂ B R ⊂ Ω a pair of concentric balls centered at x 0 ∈ Ω and we consider ξ(x) = ξ(|x − x 0 |) a smooth function such that ξ(t) = 1 for t ∈ [0, R/2] and ξ(t) = 0 for t ≥ R .
We would like to use ϕ = ξv as test function in (3.4). We first observe that, by Theorem 2.5, the function ξv belongs to C 1,D c (Ω ). Moreover, proving Lemma 2.2, we actually proved that a mollification of a Dini continuous function with compact support strongly converges in C D to the function itself. Thus, combining this fact with Lemma 2.4, we have that a ij D j (ξv) ε strongly converges in C D to a ij D j (ξv), where (ξv) ε (x) = (ρ ε * ξv)(x), ρ ε being a standard mollifier. This in turn implies that the use of ϕ = ξv as test function in (3.4) is admissible: On the regularity of very weak solutions Page 13 of 23 43 Let us come back now to the equation satisfied by v. Let u ε be a mollification of the solution u, that is u ε = ρ ε * u, with ρ ε a standard radial mollifier. We use ξu ε in (3.5): Now we claim that this implies, when we pass to the limit as ε → 0, that (3.7) Note that the most delicate terms are the two involving the gradient of u ε . For a Dini continuous function w (the domain of w is not specified since the function will be multiplied by a function with compact support) we will show that or, in other terms, recalling that μ j is the limit in the weak * topology of (C D c (Ω )) of the incremental quotient of u lim ε→0ˆD j u ε w ξ dx = μ j , w ξ .
We have: where in the last equality we used again that a mollified function of a Dini continuous function with compact support strongly converges in C D to the function itself. Thus we obtain (3.7). From it, exploiting the symmetry of a ij and using (3.6) we get (3.8) We now estimate the three terms I m , m = 1, 2, 3. We have By the definition of the norm in the space of Dini continuous functions we have By simple computation we have and, using the properties of the solution v (recall that p > n), the right hand side can be estimated as To summarize, we have The estimate of I 2 and I 3 simply comes by Hölder's inequality and again by the properties of the solution v: and At the end, the estimates proved for I 1 , I 2 and I 3 lead to Since f is an arbitrary smooth function in L p (Ω , R n ), we conclude which means, using a finite covering argument, that μ j is a function in L p loc (Ω) and then u ∈ W 1,p loc (Ω), since, for every ϕ ∈ C ∞ c (Ω) and for h small enough, we haveˆΔ j h u ϕ dx =ˆuΔ j −h ϕ dx; passing to the limit as h → 0, we derive Since u ∈ W 1,p loc (Ω), Brezis's result implies that u is a weak solution of the equation (1.5), i.e. our statement.

Sobolev coefficients
As pointed out in the Introduction, very weak solutions in L n loc (Ω) associated to coefficients in W 1,n (Ω) are not weak solutions, due to the counterexample found in [10]. The quoted references on this problem have suggested us to consider Sobolev coefficients with a modulus of continuity satisfying the double Dini condition.
On the other hand, another way to get around the counterexample is to deal with very weak solutions in L q loc (Ω), with q > n . The result is the following. Proof. The proof rests on a duality and a bootstrap argument.
Step 1 We claim that u ∈ W 1, qn q−n loc (Ω). We proceed as in the Step 1 of the proof of Theorem 1.2 to arrive to (3.2). Now we estimate the six terms I m . We use Hölder's inequality and Proposition 2.8 to get , , , , and finally, as for (3.3), .
So, arguing as in the Step 1 of Theorem 1.2, we deduce ˆΩ , which in turn implies, thanks also to Theorem 2.1, that u ∈ W 1, qn q−n loc (Ω). Let us note that thanks to this, the equation satisfied by u may be rewritten as where the test functions ϕ can be taken in W 1, qn q−n (Ω) with compact support. On the other hand, the summability of the solution u is not improved by its belonging to this Sobolev space, since qn q−n = qn qn −q+n and the Sobolev conjugate of qn qn −q+n is q.
Step 2 We prove that u ∈ W 1,q loc (Ω). As in the Step 2 of the proof of Theorem 1.2, for j ∈ {1, . . . , n} let f = (f 1 , . . . , f n ) with f j ∈ C ∞ c (Ω ) be such that j ||f j || L q (Ω ) ≤ 1. As before, we take B R/2 ⊂ B R ⊂ Ω a pair of concentric balls centered at x 0 ∈ Ω and we consider ξ(x) = ξ(|x − x 0 |) a smooth function such that ξ(t) = 1 for t ∈ [0, R/2] and ξ(t) = 0 for t ≥ R . We can choose ϕ = vξ in (4.1) and ϕ = uξ as test function in (4.2), so that We estimate the three terms I m . We have and finally where the last inequality derives from the fact that the Sobolev conjugate of q is qn q−n . To sum up we have obtained jˆf j ξD j u dx ≤ C f L q (Ω ,R n ) , as well ξDu L q (Ω ,R n ) ≤ C, and, using a finite covering argument, this implies that u ∈ W 1,q loc (Ω). Let us observe that this Sobolev regularity improves the summability of u. In particular, u ∈ L q * loc (Ω), where q * is the Sobolev conjugate of q.
Step 3 We claim that if q > n then u is a weak solution. By the previous step, we deduce that if q > n then the solution u is in L ∞ loc (Ω). At this point, it is not difficult to prove, arguing as in Step 1, that u ∈ W 1,n loc (Ω).