A regularity result for a class of elliptic equations with lower order terms

In this paper we establish the higher differentiability of solutions to the Dirichlet problem div(A(x,Du))+b(x)u(x)=finΩu=0on∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$\end{document}under a Sobolev assumption on the partial map x→A(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \rightarrow A(x, \xi )$$\end{document}. The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.


Introduction
This paper concerns the higher differentiability and the higher integrability of the gradient of local weak solution of the Dirichlet problem The operator A ∶ Ω × ℝ n → ℝ n is a Carathéodory mapping, satisfying for positive constants , > 0 the following assumptions Concerning the dependence on x-variable, it is clear that no extra differentiability can be obtained for solutions even if the data f(x) and b(x) are smooth, unless some differentiability assumption is made on the operator A(x, ).
To this aim, we shall assume that there exists a non negative function k(x) ∈ L n loc (Ω) , such that for every ∈ ℝ n and a.e. x, y ∈ Ω . Finally, we shall assume for a.e. x ∈ Ω.
Conditions (1.2) and (1.3) express the uniform ellipticity and lipschitz continuity of the operator A(x, ) with respect to the variable .
Condition (1.4), in view of the pointwise characterization of the Sobolev spaces [12], means that the partial map x → A(x, ) belongs to the Sobolev class W 1,n loc (Ω). Finally, condition (1.5), introduced in Ref. [1] (see also Ref. [2]), relates the coefficient of the lower order term with the right hand side.
This interplay yields a regularizing effect on the solution of the Dirichlet problem (1.1). More precisely, it is sufficient to assume (1.5) to obtain The key tool to deal with equations with lower order terms, assuming a low integrability for b(x) and f(x) as in (1.1), is the result in Ref. [1] (see also Ref. [2]).
Recall that the boundedness of the solution of equation (1.1) is well known if b(x), f (x) ∈ L s (Ω) for some s > n 2 [10], and usually it is the first step in the analysis of the regularity of the solutions and open the way to the investigation of some higher regularity for the solutions. We would like to mention that, under L 1 integrability on the right hand side f, existence and uniqueness of the solutions have been established in Ref. [7], when Q = 0.
Recent works have shown that the W 1,n -regularity of the map x → A(x, ) is sufficient to obtain the higher differentiability of the solutions of the problem (1.1) but in the case b = f = 0 (see Refs. [5,8,9,16,17]).
The above mentioned results hold true also for the so-called p-harmonic operator, as well as for local minimizers of integral functionals with p-growth.
Here we take advantage from the result in Ref. [2] to deal with equation as in (1.1) and prove the following Theorem 1 Let u ∈ W 1,2 0 (Ω) be the solution of the Dirichlet problem (1.1). Assume that A(x, ) satisfy (1.2)-(1.4), that b(x) ∈ L 2 (Ω) and that (1.5) is satisfied for some constant Q > 0 . Then u ∈ W 2,2 loc (Ω) and the following second order Caccioppoli type inequality holds for every ball B R ⊂ Ω.
We would like to pointout that we only require b(x), f (x) ∈ L 2 (Ω) and this assumption, which is stronger than the one used in Ref. [2], is still weaker than the classical one in order to deal with bounded solution for n ≥ 4 . On the other hand, in order to obtain higher differentiability for solutions to problem (1.1), condition (1.5) cannot be dropped without assuming higher integrability on b(x) and f(x).
Indeed, condition (1.5) is essential to have Theorem 1, under the hypothesis b(x), f (x) ∈ L 2 (see example in Section 4).
On the other hand, it is easy to check that Theorem 1 still holds true without condition (1.5), but assuming b(x) ∈ L n .
The proof of Theorem 1 is obtained combining a suitable a priori estimate for the second derivatives of the solution of the equation with an approximation argument.
To establish the a priori estimate for the solution to (1.1) which has discontinuous coefficients we use the classical tool of the difference quotient method.
The main difficulty in establishing the a priori estimate is that we need to deal with terms with critical integrability that have to be reabsorbed and we shall succeed by suitable iteration argument.
Next, we use an approximation procedure that is constructed by introducing Dirichlet problems that, on one hand, admit solutions with the second derivative in L 2 loc (Ω) and, on the other, satisfy the assumptions of the result in Ref. [1]. Also we need to prove that the solutions of each approximating Dirichlet problem are uniformly bounded by the same constant Q appearing in assumption (1.4).
As a consequence of our higher differentiability result combined with the boundedness of u we also have the following higher integrability for the gradient of the solutions Such higher integrability result is a consequence of a suitable Gagliardo Nirenberg type inequality and it is inspired by the arguments introduced in Ref. [6] (for other Gagliardo Nirenberg type inequalities see also Ref. [4]). It is well known that the W 1,n -regularity of the coefficients has been employed in the study of non variational equations [3,15]. Note that W 1,n ⊂ VMO ⊂ BMO , variational and non variational equations under VMO and BMO coefficients have been widely investigated. Among the others, we quote the results in Refs. [11,13,14,[18][19][20].
Obviously, once the existence and the W 2,2 -regularity of the solution of (1.1) has been proven, the solution u of the equation in (1.1) is also a solution to a non variational equation. In a forthcoming project we shall examine the analogous problem for the non variational case with lower order terms.

Notations and preliminary results
In Ref. [1] the authors studied boundedness of the solutions of the following problem: with 2 < p ⩽ n and A ∶ Ω × ℝ n → ℝ n is a Carathéodory function satisfying for some positive constants , and a function h(x) ∈ L p � (Ω) that and for almost every x ∈ Ω , for every and ∈ ℝ n and In Ref. [1] it has been proven that u ∈ L ∞ (Ω) . More precisely, We will use previous theorem in the case p = 2 and with h(x) = 0 = R(x) . In this particular case, the precise bound for the L ∞ -norm of u, which will be used in the sequel, is the following A regularity result for a class of elliptic equations with lower…

Difference quotient
In order to get the higher differentiability of the solutions (1.1) we will use the difference quotient method. Therefore in this section we introduce the finite difference operator and we recall the basic properties.

Definition 1 For every vector valued function
The difference quotient is defined for h ∈ ℝ⧵{0} as The following proposition describes some elementary properties of the finite difference operator and can be found, for example, Ref. [10]. The next result about finite difference operator is a kind of integral version of Lagrange Theorem.

Proposition 1 Let f and g be two functions such that
The following results will be useful in the sequel

Moreover
Now, we recall the fundamental Sobolev embedding property (for the proof we can refer, for example [10] [Lemma 8.2]).
and with c ≡ c(n, N, p).
An iteration Lemma finds an important application in the so called hole-filling method. A proof can be found, for example, Ref. [10] (see Lemma 6.1).
The following Lemma is a sort of Gagliardo Nirenberg interpolation inequality where p ∈ (1, ∞) and m > 1 . Moreover for a positive constant c = c(p).

Proof of Theorem 1
This section is devoted to the proof of our main result. We shall divide it in two steps: in the first one we establish an a priori estimate for the second derivatives of the solutions and in the second one we construct the suitable approximating problems and we shall prove that the a priori estimate is preserved in passing to the limit.

Proof Step 1. The a priori estimate
Suppose that u is a local solution of the problem (1.1) such that u ∈ W 1,2 0 (Ω) ∩ W 2,2 loc (Ω). Let us fix a ball B R ⊂ Ω and arbitrary radii R 2 < r <s < t < r < R , with ∈ (1, 2) . Consider a cut-off function ∈ C ∞ 0 (B t ) such that ≡ 1 on B̃s and |D | ≤ c t−s . Using = s,−h ( 2 s,h u) as a test function in the equation in (1.1) we get By (d1),(d2) of Proposition 1, the first integral in the left hand side in (3.1) can be written as follows:

Inserting (3.2) in (3.1)
On the other hand, the left hand side of (3.3) can be written as follows Substituting in (3.3) we obtain: Using assumptions (1.3), (1.4), (1.5) in the right hand side of the previous estimate, we get h u)).
Since the assumptions of Theorem 2 are satisfied, we can use that ‖u‖ L ∞ (Ω) ⩽ Q thus getting Using the ellipticity assumption (1.2) in the left hand side and Young's inequality in the right hand side of (3.4), since ∈ C ∞ 0 (B t ) , by Lagrange Theorem, the previous inequality gives: where we also used that |D | ⩽ C t −s .
Choosing = 4 , we can reabsorb the second integral in the right hand side of the previous estimate by the left hand side, moreover we can apply Lemma 1 and the properties of so as to obtain Dividing both sides of previous estimate by |h| 2 we obtain that In order to estimate the last integral of the previous inequality, we observe that (3.7) By the absolute continuity of the integral there exists R 0 such that

3
A regularity result for a class of elliptic equations with lower… Hence, choosing R < R 0 determined in (3.10), estimate (3.9) becomes and so Since the previous inequality is valid for all radii r <s < t < r , by iteration Lemma 3, with we deduce that: On the other hand, choosing a cut off function such that = 1 on B R 2 , ∈ C ∞ 0 (Br) , and arguing again as we did from (3.1) to (3.7) we obtain that Therefore, by estimate (3.11) we get and so, by Lemma 2, we conclude with where C = C( , , Q, n, R).
Let u ∈ W 1,2 0 (Ω) be the solution of (1.1), let us fix a ball B R ⊂ Ω � and let us denote by u ∈ W 1,2 (B R ) the unique solution of the Dirichlet problem Note that, by the classical theory, we have u ∈ W 2,2 loc (B R ) for each > 0 (see for example [10]) .
Our first aim is to prove that, for every > 0 , on B R we have To this purpose, let us set Since |u| ⩽ Q on B R then w = u = u on B R , and hence w is an admissible test function for problem (3.18), i.e.
(3.12) To this aim using u as test function in (3.18), by (3.14), we get (3.20) By using (3.19) and (3.16), we obtain Since b → b strongly in L 2 then as we wanted. The boundedness of (Du ) in W 1,2 implies that there exists v ∈ W 1,2 (B R ) such that Du ⇀ Dv in W 1,2 (B R ) and, since u = u on B R in the sense of the trace, we also have v = u on B R . Our next aim is to prove Du → Dv strongly in L 2 loc (B R ). Since u is a solution of (3.18), it is well known that u ∈ W 2,2 loc (B R ) , and, since |f | < Qb and ‖u ‖ ∞ ⩽ Q , we can apply the a priori estimate in (3.12) to each u , thus getting for every B r , r < R . This implies u ⇀ v weakly in W 2,2 loc (B R ) and then u → v strongly in W 1,2 loc (B R ). Now we want to show that v is local solution of the equation in (1.1) . Since u is a solution of (3.18), we have On the other hand, by (3.14) it holds that i.e.
Definitively we proved that u ⇀ u in W 2,2 loc (B R ) then u → u strongly in W 1,2 loc (B R ) , hence we can conclude the proof, simply by passing to limit in (3.12), since the operator A (x, ) satisfies (3.14)- (3.17). ◻ Proof of Corollary 1. It is sufficient to combine inequality in Theorem 1 with Lemma 4 for p = 2 contained in Ref. [9].
Actually b(x) and f(x) belong to L p for all p < n + 2 , nevertheless u(x) ∉ W 2,2 , hence Theorem 1 is not true.