Abstract
We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form \(\partial _{\alpha }(A_{ij}^{\alpha \beta } \partial _{\beta } u^{j}) = 0\). It is known that the Dini continuity of coefficient matrix \(A = (A_{ij}^{\alpha \beta }) \) is essential for the differentiability of solutions. We prove the following results:
(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,2 of A satisfies
where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.
(b) If XA,2 = 0, then ∇u ∈ V MO.
(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when \(\nabla u \in L^{\infty } \cap VMO\).
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1 Introduction
Let n ≥ 2, N ≥ 1 and consider the elliptic system for u = (u1,…,uN)
where B4 is the ball in \(\mathbb R^{n}\) of radius four and centered at the origin, and the coefficient matrix \(A = (A_{ij}^{\alpha \beta })\) is assumed to be bounded and measurable in \(\bar B_{4}\) and to satisfy, for some positive constants λ and Λ,
It is well known that if the coefficient matrix A belongs to \(C^{0,\alpha }_{\text {loc}}(B_{4})\) then every solution u ∈ H1(B4) of (1.1) belongs to \(C^{1,\alpha }_{\text {loc}}(B_{2})\); see, e.g., Giaquinta [13, Theorem 3.2] where the result is attributed to Campanato [7] and Morrey [24]. It was conjectured by Serrin [26] that the assumption u ∈ H1(B4) can be relaxed to u ∈ W1,1(B4). This has been settled in the affirmative by Brezis [2, 3]. (See Hager and Ross [14] for the relaxation from u ∈ H1(B4) to u ∈ W1,p(B4) for some 1 < p < 2.) Moreover, in [2, 3], it was shown that if A satisfies the Dini condition
then every solution u ∈ W1,1(B4) of (1.1) belongs to C1(B2). For related works on the differentiability of weak solutions under suitable conditions on \(\bar \omega _{A}\), see also [15, 22, 23].
Differentiability of weak solutions under weaker Dini conditions involving integral mean oscillation of A has also been studied. For 0 < r ≤ 2, let
In Li [21] it was shown that if
then every solution u ∈ H1(B4) of (1.1) belongs to C1(B2). In Dong and Kim [12] (see also [9]), this conclusion was shown to remain valid under the weaker condition that
(Note that the finiteness of \({{\int \limits }_{0}^{2}} \frac {\omega _{A}(t)}{t} dt\) or \({{\int \limits }_{0}^{2}} \frac {\bar \varphi _{A}(t)}{t} dt\) implies that A is continuous.) For related works for nonlinear elliptic equations, see, e.g., [19, 25, 28] and the references therein.
The Dini condition (1.4) and its integral variants (1.5) and (1.6) are phenomenologically sharp for the differentiablity of weak solutions of (1.1). In Jin, Maz’ya and van Schaftingen [17], examples of continuous coefficient matrices A with moduli of continuity \(\bar \omega _{A}(t) \sim \frac {1}{|\ln t|}\) as \(t \rightarrow 0\) were given showing the following phenomena:
∙ There exists a solution u ∈ W1,1(B4) of (1.1) such that u ∈ W1,p(B4) for all \(p \in [1,\infty )\), and ∇u ∈ BMOloc(B4) but \(\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})\) and ∇u∉V MOloc(B2)Footnote 1.
∙ There exists a solution u ∈ W1,1(B4) of (1.1) such that u ∈ W1,p(B4) for all \(p \in [1,\infty )\) but ∇u∉BMOloc(B2).
In this paper, we consider mean oscillation estimates for ∇u when A slightly fails the Dini conditions (1.4), (1.5) and (1.6). For \(1 \leq p < \infty \), let \(\omega _{A,p}: (0, 2] \rightarrow [0,\infty )\) denote the Lp mean oscillation of A
It is clear that ωA,1 = ωA, \(\omega _{A,2} \leq \bar \varphi _{A}\), ωA,p is non-decreasing in p, and \(\omega _{A,p} \leq \bar \omega _{A}\) for all \(p \in [1,\infty )\).
We now state our first result.
Theorem 1.1
Let \(A = (A_{ij}^{\alpha \beta })\) satisfy (1.2) and (1.3). There exists a constant C∗ > 0, depending only on n, N, Λ and λ such that if
then every solution u ∈ H1(B4) of (1.1) satisfies ∇u ∈ BMOloc(B2). Moreover, if
then every solution u ∈ H1(B4) of (1.1) satisfies ∇u ∈ V MOloc(B2).
Note that condition (1.7) implies that \(\omega _{A,2}(t) \rightarrow 0\) as \(t \rightarrow 0\), i.e., A ∈ V MOloc(B2).
Remark 1.2
Let \(1 < p < \infty \). Theorem 1.1 remains valid if ωA,2 is replaced by ωA,p and the regularity assumption u ∈ H1(B4) is replaced by u ∈ W1,p(B4), where the constant C∗ is now allowed to depend also on p. For p ≥ 2, this follows from the inequality ωA,2 ≤ ωA,p for those p. For 1 < p < 2, see Proposition 3.2.
It is clear that if ωA,2 satisfies (1.5), then it satisfies (1.8) (and hence (1.7)). The following lemma gives examples which satisfy (1.8) but not necessarily (1.5).
Lemma 1.3
If \(\limsup _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} < \frac {1}{C_{*}}\), then XA,2 = 0. If \(\liminf _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t}\) \( > \frac {1}{C_{*}}\), then \(X_{A,2} = \infty \).
We note that, in case \(\omega _{A,2}(t) \ln \frac {1}{t} \rightarrow 0\) as \(t \rightarrow 0\), the BMO regularity of ∇u was proved by Acquistapace [1]. (See also [16].)
By Lemma 1.3, an explicit example of ωA,2 satisfying (1.8) (for any constant C∗) but not (1.5) is
In addition, unlike (1.5) or (1.6), (1.8) does not imply that A is continuous, e.g.,
(This can be checked using the fact that the function \(s \mapsto {\sin \limits } s\) is Lipschitz on \(\mathbb R\) and the fact that the function \(x \mapsto \ell (x) := \ln \ln \ln \frac {64}{|x|}\) has L2 mean oscillation \(\omega _{\ell ,2}(t) \sim \frac {1}{\ln \frac {64}{t} \ln \ln \frac {64}{t}}\).)
When A is merely of vanishing mean oscillation, we note that
Remark 1.4
Let \(A = (A_{ij}^{\alpha \beta })\) belong to V MO(B4) and satisfy (1.2) and (1.3). Then every solution \(u \in W^{1,\infty }(B_{4})\) of (1.1) satisfies ∇u ∈ V MO(B2). See [12, equation (2.14)]. See also Section 2 for a different proof.
The obtained regularity in Theorem 1.1 and Remark 1.4 appears sharp. As in [17], counterexamples can be produced to show that, under (1.8),
∙ Solutions of (1.1) may not have bounded gradients (though their gradients are of vanishing mean oscillation by Theorem 1.1),
∙ \(W^{1,\infty }\) solutions of (1.1) may not be differentiable (though their gradients are of vanishing mean oscillation by Remark 1.4).
Proposition 1.5
There exist a coefficient matrix \(A = (A_{ij}^{\alpha \beta }) \in C(\bar B_{4})\) satisfying (1.2), (1.3) and (1.8) and a solution u ∈ H1(B4) of (1.1) such that ∇u ∈ V MO(B4) but \(\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})\).
Proposition 1.6
There exist a coefficient matrix \(A = (A_{ij}^{\alpha \beta })\in C(\bar B_{4})\) satisfying (1.2), (1.3) and (1.8) and a solution u ∈ H1(B4) of (1.1) such that \(\nabla u \in L^{\infty }(B_{4}) \cap VMO(B_{4})\) but ∇u∉C(B2).
Theorem 1.1 is a consequence of the following proposition on the mean oscillation of the gradient ∇u in terms of the L2 mean oscillation ωA,2 of A.
Proposition 1.7
Let \(A= (A_{ij}^{\alpha \beta })\) satisfy (1.2) and (1.3). Then there exists a constant C∗ > 0, depending only on n, N, Λ and λ such that for every u ∈ H1(B4) satisfying (1.1) and for 0 < r ≤ R/4 ≤ 1/2, there hold
and
where \((\nabla u)_{r} = \frac {1}{|B_{r}|} {\int \limits }_{B_{r}} \nabla u dx\) for 0 < r ≤ 2.
Moreover, if \(u \in W^{1,\infty }(B_{4})\), then, for 0 < r ≤ R/4 ≤ 1/2,
Remark 1.8
Let 1 < p < 2. Under an additional assumption that \([A]_{BMO(B_{4})}\) is sufficiently small, the estimates in Proposition 1.7 hold if ωA,2 is replaced by ωA,p and the regularity assumption u ∈ H1(B4) is replaced by u ∈ W1,p(B4). We do not know if this smallness assumption can be dropped except for p close to 2. See Proposition 3.2.
Remark 1.9
It would be interesting to see if Theorem 1.1 and Proposition 1.7 remain valid if ωA,2 is replaced by ωA,1. In view of [9, 12], it is plausible that the answer is affirmative, but this is not clear from the techniques used in the present paper.
Remark 1.10
In the above, to keep things simple, we chose to state our results for the homogeneous system (1.1) without lower order terms. They can be generalized for non-homogeneous systems or to allow for lower order terms. See, e.g., Proposition 3.1.
2 Proofs of the Main Results
Proof of Lemma 1.3
We claim: For δ ∈ (0,1) and \(a \in (0,\infty )\), the limit
satisfies \(L_{a} = \infty \) if a > 1, La = 1 if a = 1 and \(L_{a} \leq (\ln \frac {1}{\delta })^{a-1}\) if a < 1.
When a = 1, the claim is clear. By integrating by parts, we have
If a < 1, we see from (2.1) that
To prove the claim in the case a > 1, we may assume without loss of generality that a < 2. Note that (2.1) implies
As La− 1 is finite (as 1 < a < 2), we thus have that \(L_{a} = \infty \). The claim is proved.
We now apply the claim to obtain the desired conclusions. Consider first the case that \(\limsup _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} < \frac {1}{C_{*}}\). Then there exist \(\varepsilon \in (0,\frac {1}{C_{*}})\) and δ ∈ (0,1) so that \(\omega _{A,2}(t) \leq \varepsilon (\ln \frac {1}{t})^{-1}\) in (0,δ). For \(\hat \delta \in (0,\delta )\), we compute
As C∗ε < 1, we can apply the claim to obtain
Sending \(\hat \delta \rightarrow 0\), we obtain that XA,2 = 0.
Consider next the case that \(\liminf _{t \rightarrow 0} \omega _{A,2}(t) \ln \frac {1}{t} > \frac {1}{C_{*}}\). Then there exist \(b > \frac {1}{C_{*}}\) and δ ∈ (0,1) so that \(\omega _{A,2}(t) \geq b (\ln \frac {1}{t})^{-1}\) in (0,δ). We then have
As C∗b > 1, we deduce from the claim that \(X_{A,2} = \infty \) as desired. □
Proofs of Theorem 1.1 and Remark 1.4
The results follow immediately from Proposition 1.7. □
In order to prove Proposition 1.7, we need the following estimate for harmonic replacements. (Compare [5, Lemma 3.5], [20, Lemma 3.1].)
Lemma 2.1
Let \(A, \bar A\) satisfy (1.2) and (1.3) with \(\bar A\) being constant in B4 and \(f = (f_{i}^{\alpha }) \in L^{2}(B_{4})\). Let R ∈ (0,2) and suppose u,h ∈ H1(B2R) satisfy
Then there exists a constant C > 0 depending only on n,N,Λ and λ such that
Proof
In the proof, C denotes a generic positive constant which depends only on n, N, Λ and λ. Using that \(\bar A\) is constant, we have by standard elliptic estimates that
Observing that
we deduce that
To estimate \(\|u - h\|_{L^{1}(B_{2R})}\), fix some t > 0 and consider an auxiliary equation
Testing the above against u − h, we obtain
As u − h satisfies
we have
Inserting (2.4) into (2.3) and noting that \(\|\nabla \phi \|_{L^{\infty }(B_{2R})} \leq CR\) (as \(|\partial _{\beta }(\bar A_{ij}^{\alpha \beta } \partial _{\alpha } \phi ^{i}) | \leq 1\)), we arrive at
Noting that the constant C is independent of t, we may send \(t \rightarrow 0\) to obtain
The conclusion follows from (2.2) and (2.5). □
Proof of Proposition 1.7
We only need to give the proof for a fixed R, say R = 2. Our proof is inspired by that of [21].
In the proof, C denotes a generic positive constant which depends only on n, N, Λ and λ. In particular it is independent of the parameter k which will appear below. Also, we will simply write ω instead of ωA,2.
Proof of (1.9): For k ≥ 0, let Rk = 4−k, \(\bar A_{k} = (A)_{B_{2R_{k}}}\) and \(h_{k} \in H^{1}(B_{2R_{k}})\) be the solution to
Let \(a_{k} = R_{k}^{-n/2}\|\nabla (u - h_{k})\|_{L^{2}(B_{R_{k}})}\) and \(b_{k} = \|\nabla h_{k}\|_{L^{\infty }(B_{R_{k}})}\).
Note that, by triangle inequality, we have
By elliptic estimates for hk, we have
By Lemma 2.1,
Hence
Next, we have by (2.10) that
Note that hk+ 1 − hk satisfies
As \(\bar A_{k}\) is constant and
we have by standard estimates for elliptic equations with constant coefficients and (2.8) and (2.9) (applied to hk+ 1) that
By (2.12),
We deduce that
where
and where we have used the fact that ω(t) ≤ Cω(s) whenever 0 < t ≤ s ≤ 4t. We have thus shown that
Estimate (1.9) is readily seen.
Proof of (1.10): We write
Using the estimate \(\|\nabla ^{2} h_{R_{0}}\|_{L^{\infty }(B_{1})} \leq C\|\nabla u\|_{L^{2}(B_{2})}\) together with (2.13) and (2.15), we have
where we have again used the fact that ω(t) ≤ Cω(s) whenever 0 < t ≤ s ≤ 4t. This implies
Combining (2.17) with (2.10) and (2.15), we get
As ω(2Rk) ≤ Cω(t) whenever 2Rk ≤ t ≤ 4Rk, we have
Using this in (2.18), we deduce that for k ≥ 1 that
Estimate (1.10) follows.
Proof of (1.11): We adjust the proof of (1.10) exploiting the fact that \(\nabla u \in L^{\infty }(B_{2})\). First, using the fact that \(a_{k} + b_{k} \leq C \|\nabla u\|_{L^{\infty }(B_{2})}\) in (2.13) we get instead of (2.16) the stronger estimate
and so
Combining (2.19) with (2.10), we get for k ≥ 1 that
Estimate (1.11) follows. □
Remark 2.2
If the Dini condition (1.4) or (1.5) holds, it can be seen from (2.12) that {∇hk(0)} converges to some \(P \in \mathbb R^{N \times n}\), from which it follows that
yielding the continuity of ∇u at the origin. We have thus recovered the results on the continuous differentiability of H1 solutions of Brezis [2, 3] and Li [21].
Proof of Proposition 1.5
We take N = 1 and drop the indices i, j in the expression of A (so that A = (Aαβ)). Following [17, Lemma 2.1], we make the ansatz that
Then
Selecting now
we see that A is continuous in \(\bar B_{4}\), satisfies (1.2) and (1.3) and u is an H1 solution of (1.1). The matrix A admits a modulus of continuity \(\bar \omega _{A}(t) \sim \frac {1}{\ln \frac {64}{t} \ln \ln \frac {64}{t}}\) as \(t \rightarrow 0\) and so (1.8) holds. It is readily seen that u ∈ W1,p(B4) for all \(p \in [1,\infty )\), ∇u ∈ V MO(B4) but \(\nabla u \notin L^{\infty }_{\text {loc}}(B_{2})\). □
Proof of Proposition 1.6
Instead of the choice in the proof of Proposition 1.5, we now choose
It is readily checked that A is continuous in \(\bar B_{4}\), satisfies (1.2), (1.3) and (1.8) and u is an H1 solution of (1.1), \(\nabla u \in L^{\infty }(B_{4}) \cap VMO(B_{4})\) but ∇u∉C(B2). □
3 Some Extensions
As announced in the introduction, Proposition 1.7 can be adapted for non-homogeneous systems with or without lower order terms. To illustrate, consider for example the system
Let \(\omega _{f,2}:(0,2] \rightarrow [0,\infty )\) denote the L2 mean oscillation of f
We have
Proposition 3.1
Let \(A= (A_{ij}^{\alpha \beta })\) satisfy (1.2) and (1.3). Then there exists a constant C∗ > 0, depending only on n, N, Λ and λ such that for every u ∈ H1(B4) satisfying (3.1) with f ∈ L2(B4) and for 0 < r ≤ R/4 ≤ 1/2, there hold
and
Moreover, if \(u \in W^{1,\infty }(B_{4})\), then, for 0 < r ≤ R/4 ≤ 1/2,
Proof of Proposition 3.1
The proof is a simple adjustment of that of Proposition 1.7. We will only point out the main changes.
Proof of (3.2): Define hk, ak, bk as in the proof of Proposition 1.7. Proceeding as before but paying attention to the application of Lemma 2.1 which now involves an inhomogeneous right hand side, we consecutively obtain the following estimates:
Hence, instead of (2.15), we now get
where we recall the notation \({\varOmega }(t) = {{\int \limits }_{t}^{2}} \frac {\omega (s)}{s} ds\). Estimate (3.2) follows.
Proof of (3.3): Instead of (2.16) and (2.17), we now have
and
It then follows that
Noting that, on the right hand side, the third term can be absorbed into the first term, and the fourth and fifth terms can be absorbed into the second term, we arrive at estimate (3.3).
Proof of (3.4): Using \(\nabla u \in L^{\infty }(B_{2})\), we obtain this time that
and so
It then follows that
Estimate (3.4) is readily seen. □
Finally, we briefly touch on the validity of Theorem 1.1 when ωA,2 is replaced by ωA,p for 1 < p < 2. For this, we only need the following Lp version of Proposition 1.7.
Proposition 3.2
Let \(A= (A_{ij}^{\alpha \beta })\) satisfy (1.2) and (1.3). Let 1 < p < 2. Then there exist constants γ > 0 and C∗ > 0 depending only on n, N, p, Λ and λ such that, provided [A]BMO(B4) < γ, there hold for every u ∈ W1,p(B4) satisfying (1.1) and for 0 < r ≤ R/4 ≤ 1/2 that
and
where \((\nabla u)_{r} = \frac {1}{|B_{r}|} {\int \limits }_{B_{r}} \nabla u dx\) for 0 < r ≤ 2.
Moreover, if \(u \in W^{1,\infty }(B_{4})\), then, for 0 < r ≤ R/4 ≤ 1/2,
The proof of Proposition 3.2 is the same as that of Proposition 1.7, but now using the following harmonic replacement estimate:
Lemma 3.3
Let 1 < p < 2. Let \(A, \bar A\) satisfy (1.2) and (1.3) with \(\bar A\) being constant in B4 and \(f = (f_{i}^{\alpha }) \in L^{p^{\prime }}(B_{4})\) with \(p^{\prime } = \frac {p}{p-1}\). Let R ∈ (0,1) and suppose u,h ∈ W1,p(B4R) satisfy
Then there exist constants γ > 0 and C > 0 depending only on n,N,p,Λ and λ such that, provided \([A]_{BMO(B_{4R})} \leq \gamma \),
Proof
We amend the proof of Lemma 2.1 using Lp theories for elliptic systems whose leading coefficients have small BMO semi-norm.Footnote 2 In the proof, C denotes a generic positive constant which depends only on n, N, p, Λ and λ.
It is known that (see, e.g., Dong and Kim [10, 11])Footnote 3, provided \([A]_{BMO(B_{4R})} \leq \gamma \) for some small enough γ depending only on n,N,p,Λ and λ, one has
Using that \(\bar A\) is constant, we have by standard elliptic estimates that
Using
and once again the fact that \([A]_{BMO(B_{4R})} \leq \gamma \), we have
To estimate \(\|u - h\|_{L^{1}(B_{2R})}\), recall from the proof of Lemma 2.1 the chain of identities
which imply
Noting that the constant C is independent of t, we may send \(t \rightarrow 0\) to obtain
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The author would like to thank Professor Yanyan Li for drawing his attention to the problem. The author would also like to thank the referee for useful comments.
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Dedicated to Professor Duong Minh Duc on the occasion of his 70th birthday
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Nguyen, L. Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients. Acta Math Vietnam 48, 117–132 (2023). https://doi.org/10.1007/s40306-022-00493-y
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DOI: https://doi.org/10.1007/s40306-022-00493-y