Counterexamples to maximal regularity for operators in divergence form

In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$.


Introduction
Let V and H be complex Hilbert spaces such that V ֒→ H densely and continuously.Identifying H * with its dual, we can view H as a subspace of V * , which is the dual of V .We start with the abstract problem (CP) u ′ − Au = f, on (0, 1), u(0) = 0.
Under what condition on A does the following hold: for all f ∈ L p (0, 1; V * ) there is a unique weak solution u which is in L p (0, 1; V )?Using the equation this also gives u ∈ H 1,p (0, 1; V * ).Problem 1 can equivalently be formulated to the question whether there is a constant C > 0 such that for all step functions f valued in H and with u the unique solution to (CP) given by Lions' result for p = 2 one has the estimate u L p (0,1;V ) ≤ C f L p (0,1;V * ) .
It is well-known that regularity results fail for the endpoint cases p = 1 and p = ∞ unless V = V * and thus A is a family of bounded operators on V (see [6,13] and also [15,Thm. 17.4.4 & Cor. 17.4.5]).Hence, we can safely concentrate on the case p ∈ (1, ∞) in this paper.
The second problem is a variation of Lions' problem [18, p. 68], who originally asked this question for symmetric A.

Problem 2.
Under what condition on A does the following hold: for all f ∈ L 2 (0, 1; H) the unique weak solution u satisfies u ′ ∈ L 2 (0, 1; H)?
Again using the equation one also has Au ∈ L 2 (0, 1; H).However, it is unclear what this tells about the regularity of u since the domain of the operators A(t) in H are not easy to describe in general.
Both problems have in common (at least if p > 2) that they imply regularity of u such as u ∈ C α ([0, 1]; [H, V ] λ ) for some α > 0 and λ ∈ (0, 1).Such properties are for instance useful in the study of non-linear problems.They follow from standard interpolation estimates and Sobolev embeddings.In Lions' general L 2 -setting one can only obtain C([0, 1]; H) or Hölder regularity of small order, compare with Theorem 2.2.
If A is autonomous, that is to say, the family A(t) does not depend on t, then in both problems the answer is affirmative.Indeed, in Problem 1 this can be concluded from Lions' result and [15,Thm. 17.2.31].Concerning Problem 2, this follows from a result of de Simon [21].
Otherwise, some conditions are needed.In the case of Problem 1 it is sufficient that the mapping t → A(t) ∈ L(V, V * ) is (piecewise relatively) continuous [2,20].Without any continuity it is only known that there is some ε > 0 depending on Λ, λ such that Problem 1 holds with p ∈ (2 − ε, 2 + ε), see Theorem 2.2.For Problem 2 the situation is even more delicate.Based on an abstract counterexample for the Kato square root problem due to McIntosh, Dier constructed a first nonsymmetric counterexample to Problem 2 in his PhD thesis, see also [3,Sec. 5].Moreover, Fackler constructed a counterexample that is symmetric and C 1 2 -Hölder continuous [11].To the contrary, with slightly more regularity (for instance C 1 2 +ε -Hölder regularity), many positive results were given [1,4,7,9,12,14,20].We also recommend the survey [3] for an overview of Problem 2.
The counterexamples due to Dier and Fackler are abstract and not differential operators.As is highlighted by the solution to the Kato square root problem [5], the extra structure of a differential operator can be beneficial compared to the general situation.It is hence of interest to find counterexamples to Problems 1 and 2 that are differential operators.For Problem 2 this was explicitly pointed out in [12,Problem 6.1] and [3, Prop.12.1].To be more precise with our setting, we work with In the notation of (CP) we have put A(t)v = −div(B(t)∇v).Lions' theory yields a unique solution u in the regularity class . Problems 1 and 2 ask if for all elliptic coefficients one has the regularity u ∈ L p (0, 1; ).The answer is negative in both cases and the respective counterexamples will be the content of the present article.More precisely, based on a construction of the second-named author [19], we will obtain equations whose solution fails to have higher L r (L s )integrability (Theorem 2.3).Based on this result, we produce a counterexample to Problem 1 for every p = 2 in Theorem 2.4.In particular, this shows sharpness of Theorem 2.2 which we mentioned briefly above.Concerning Problem 2, we will show in Theorem 2.6 that it fails in the worst possible way for general elliptic problems in divergence form.

Maximal regularity in Hilbert spaces
2.1.Known results.We present the results regarding Problem 1 that were already discussed in the introduction.
where the integrals are defined as Bochner integrals in V * .
One can check that u is a solution to (CP) if and only if for all φ ∈ C ∞ c ((0, 1); V ) one has Theorem 2.1 (Lions).In the situation of (CP) there is a unique solution u ∈ L 2 (0, 1; V ) ∩ H 1 (0, 1; V * ).Constants in the maximal regularity estimate depend only on Λ and λ.
Based on a perturbation principle for isomorphisms in complex interpolation scales due to Sneiberg [22], Lions' result can be extended to p close to 2, see [10,Thm. 4.2].
It will be clear from Theorem 2.4 that one cannot improve the latter to all p ∈ (1, ∞), even if f ∈ L p (0, 1; H).

2.2.
Failure of L r (L s )-integrability for variational solutions.The following counterexample is based on a construction due to the second-named author [19] and is the basis for our subsequent counterexamples to Problems 1 and 2.
Step 1: general setup.We repeat part of the construction in [19].There exist a Lipschitz function w : R d → C and an elliptic matrix a : R d → C d×d with the subsequent properties.The functions a and w are uniformly bounded, smooth away from zero, Lipschitz in zero, and for all multiindices α, |x| ≥ 1, These estimates follow from the definitions of w and a given in [19,Sec. 3].Similar as in [19, Eq. ( 5), Thm 2.2, and Cor 2.3] define These were constructed so that ) and the equation is satisfied pointwise for t ∈ (0, 1) and x ∈ R d \ {0}.The latter means that It is clear from the constructions that the weak derivative ∂ t ζ exists on (0, T ) as an L 2 (B R )-valued function, for any T < 1 and R > 0. Now let )) and u is the unique solution to (P).Then, using (2.4) it is elementary to check that for t ∈ (0, 1) and x ∈ R d \ {0} one pointwise has Moreover, it also holds in distributional sense on (0, 1) × R d .Hence, by density and (2.1), it is a solution to (P).
Therefore, it remains to check u ∈ L 2 (0, 1; ) in order to find that u is the unique solution to (P) provided by Theorem 2.1.
Step 2: u ∈ L 2 (0, 1; H 1 (R d )).Fix t ∈ (0, 1) and let α be a multiindex with |α| ≤ 1.By definition of ζ and using (2.2) one has for some constant C ′ d,w only depending on d and w and with Note that C d,w is finite since w is Lipschitz.Recall that d 4 − µ 2 is positive.Thus, by definition of u and using (2.5) we find Thus, u is bounded as an L 2 (R d )-valued function and therefore in particular u ∈ L 2 (0, 1; L 2 (R d )).Similarly, By choice of µ we have To estimate the norm of f we consider all parts separately.The fact that ηζ ∈ L ∞ (0, 1; L 2 (R d )) follows from (2.5).Due to the support properties of ∇η and the boundedness of B it suffices to prove a uniform estimate for It remains to estimate ζdiv(B∇η).Again, due to the support properties of ∇η, it suffices to consider x ∈ B 2 \ B 1 .First, by (2.2) and definition of ζ, Thus it remains to estimate Since B k,ℓ and ∂ β η are uniformly bounded, it is enough to show that is uniformly bounded for |α| = 1.Indeed, this follows from the analogous calculation to (2.6) using (2.3) instead of (2.2).

2.3.
Negative result concerning problem 1.The following result shows that for time-dependent operators in the variational setting, maximal L 2 -regularity cannot be extrapolated to maximal L p -regularity for p = 2 besides the small interval given in Theorem 2.2.It answers Problem 1 in a negative way in the setting of elliptic differential operators.
Proof.We divide the proof into two cases.Case 1: p > 2. We appeal to Theorem 2.3.If d ≥ 3 put r = p and s = 2 * := 2d d−2 and if d = 2 put r = p and s > 2p p−2 .In both cases the condition . This implies u ∈ L p (0, 1; H 1 (R d )) since otherwise the Sobolev embedding yields a contradiction to the previous assertion.
Case 2: p < 2. We use a duality argument that resembles [16,Thm. 6.2].By the first part there are f ∈ L p ′ (0, 1; . Assume for the sake of contradiction that for every satisfying the estimate where the constant C > 0 does not depend on v and g.By translation the question on (−1, 0) is equivalent to that on (0, 1).Both intervals are related by the transformation v → −v.We write for instance u(t) = u(−t) to translate u to a function on (−1, 0) and vice versa.Specialize A = ( B) * in (2.8).Since u(0) = 0 = v(1) we can use integration by parts to obtain Hence, using (2.9) we can estimate Since g was arbitrary, deduce by duality the contradiction u ∈ L p ′ (0, 1; H 1 (R d )).

2.4.
Negative result concerning problem 2. We show that Problem 2 fails in the worst possible way in the case of elliptic operators in divergence form: for every ν ∈ (1/2, 1] there exist coefficients B and a forcing term f ∈ L ∞ (0, 1; L 2 (R d )) such that the unique solution u to (P) satisfies u / ∈ H ν (0, 1; L 2 (R d )).This solves Lions' problem in a negative way also for elliptic operators.It would still be interesting to find a counterexample for ν = 1 where the coefficients are Hölder continuous of order α for some α ≤ 1/2, and this would then be an optimal counterexample due to the positive results for which we refer to the survey [3].The present coefficient function cannot even be continuous, see Remark 2.7.There seems to be some room for improvement in the regularity of B as the regularity is much worse than H 1 in time.
The examples show some limitations of what regularity estimates can hold for elliptic operators in divergence form.However, several issues remain for Problems 1 and 2. For instance, the operator A used in the above counterexamples is not symmetric/hermitian.We do not know what can be said for the case d = 1.The only reference we found on the one-dimensional setting is [17], where counterexamples are given in case of L p -integrability in time and space for both the divergence and non-divergence setting for the range p ∈ (1, 3/2) ∪ (3, ∞).Finally, it would be interesting to see what can be said about Problem 2 if A is more regular in time (i.e.continuous or Hölder continuous).