1 Introduction

Optimal estimates from below for biharmonic Green functions

Abstract.

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary -smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.

2000 Mathematics Subject Classification:
Primary 35B51; Secondary 35J40, 35A08

1. Introduction

It is well-known that the Green function for second order elliptic equations on bounded domains can be estimated from above and from below by positive multiples of the same (positive) function, which is explicitly given in terms of the distances to the boundary, , and the distance . See for example [3]. The behaviour of the biharmonic Green function for Dirichlet boundary conditions should be somehow similar but will have two crucial distinctions. The singularity of course is of lower order, namely instead of with the dimension, but a more serious distinction is the fact that the biharmonic Green function is not everywhere positive for most domains. Indeed, the few known domains with a biharmonic Green function of a fixed positive sign are balls in arbitrary dimensions, small perturbations of those balls and of some limaçons in dimensions. See respectively [2, 11, 10] and [6, 7]. The results in [7] extend and correct [13].

It has been observed numerically on domains with a sign changing biharmonic Green function that the negative part is rather small and that it is also not located near the singularity. The aim of this paper is to give optimal estimates from below. Previous results concerning smallness of the negative part have been obtained in [10] for and [4] for . With the estimates for the absolute value of that Green function in [5] these previously known estimates are, when , as follows:

(1)

for all and where are some positive constants only depending on the domain. The distance of to the boundary is defined by

and denotes the said Green function. Let us remind the reader that the biharmonic Green function is such that

solves

(2)

In (1) the dimension is restricted to . As has been shown in [10] for and in [4] for , the estimate from below in (1) holds in all dimensions. An estimate from above has also been proved for but the formula for that estimate is different from (1). Those estimates can be found in [5] and contain the function in (4).

The main result is the estimate from below in the following theorem. For the sake of completeness we include the estimate from above.

Theorem 1.

Let be a bounded -smooth domain. Let denote the biharmonic Green function in for (2). Then there exist constants , , depending on the domain , such that we have the following Green function estimate:

(3)

for all , where

(4)
Figure 1. The dashed curve gives the typical behaviour of ; the shaded area describes the band given by .

The estimate from above follows from [5, 15, 16] so that only the estimate from below has to be proved here. One should observe that was proved for some suitable in [10]. The preceding result may be considered as an extension of the estimates in [10] showing that close to the pole the positive singular behaviour of the fundamental solution can also be seen in the Green function.

Due to the different behaviour of the Green function we need to distinguish between and in proving Theorem 1. Finally we should remark that the lack of a maximum principle not only results in sign changing Green functions but also complicates the proof of these estimates in the fourth order case. The proof in the second order case heavily depends on the Harnack inequality which in turn depends on the maximum principle.

An interesting consequence of Theorem 1 is a uniform local positivity result. When this was proved in [10], while for we refer to [9, Theorem 6.15]. Here the emphasis is on the interplay between Theorem 1 and the following result. Moreover, we provide a proof for which is much simpler than the previous one and in the same spirit as for .

Theorem 2.

Let be a bounded -smooth domain. Let denote the biharmonic Green function in for (2). Then there exist a constant , such that

(5)

2. Some auxiliary results for

A careful inspection of the proofs in Nehari [17] and Grunau-Sweers [12] shows the following local estimate for the biharmonic Green function from below.

Proposition 3.

Let . Then there exists constants and , which depend only on the dimension , such that the following holds true.

Assume to be a -smooth bounded domain and let denote the Green function for the biharmonic operator under Dirichlet boundary conditions. If

(6)

then we have

For the constant , one may achieve that .

In dimension it seems impossible to achieve a global linear dependence of the radius of a ball of guaranteed positivity on the boundary distance, see Lemma 7 and [17]. In other words, the best result seems , which strictly depends on .

Due to Proposition 3 and using the same constants as there we may restrict ourselves in what follows to such that

(7)
Lemma 4.

Suppose that and that is a bounded -smooth domain. Then for each there exists a radius and a constant such that for all subject to condition (7) one has

(8)
Proof.

We only need to discuss since for interior points one may choose so small that condition (7) becomes void.

We assume by contradiction that there exist subject to (7) such that

(9)

In particular we have , , . Without loss of generality we may assume that and that the first unit vector is the exterior unit normal to at .

Figure 2. and subdomain for .

For large enough, we may define as the closest boundary point to . We introduce the rescaled biharmonic Green functions

for

Since , the exterior unit normal at converges to the first unit vector and so we conclude that

It was proved in [10, Lemma 7] that locally uniformly in

where and is the dimensional volume of . We remark that this step and in particular the required uniqueness proof for were carried out in Grunau-Robert [10] using the assumption . The necessary modifications for are emphasized in the proofs of Lemmata 7 and 6. Assumption (9) gives

(10)

where

After passing to a further subsequence we find with , . In view of the local smooth convergence of to the biharmonic Green function in the half space , Boggio’s formula and there exists a positive constant such that for large enough:

This contradicts (10) and the proof of the lemma is complete. ∎

3. Proof of the main estimate for

Supposing that (6) holds, i.e. , one finds that even and hence

So in that case the estimate from below in Theorem 1 follows directly from Proposition 3. Hence, we may assume from now on, again, that are subject to condition (7). Applying a compactness argument to

we see that there exist positive numbers , , such that implies that . If we take from [5], cf. also [9], that so that

Since we end up with

and positive constants in this case. The proof of Theorem 1 for is complete.

4. Auxiliary results for

Lemma 5.

Let and let be a biharmonic Green function with Dirichlet boundary condition, that is

for all and . Moreover, we assume that for all and that a growth condition holds at infinity:

(11)

Then is uniquely determined and given by Boggio’s formula [2]:

with .

Proof.

We choose some arbitrary and keep it fixed in what follows. We write

where is a regular function in and in , for . According to [8, 14], or by checking directly, with ,

satisfies and is biharmonic on . Since satisfies (11), so does . So we have

Using local elliptic estimates and their scaling behaviour for biharmonic functions satisfying Dirichlet boundary conditions on , that is,

we find that

Having these estimates for and these first two derivatives, we have an estimate for and may repeat the above arguments to find similar estimates for the derivatives of and even

The maximum principle applied to the harmonic function shows that

Letting yields

with a suitable function . This shows that any is harmonic and, as shown above, as . Hence, any and by Taylor’s formula and observing the boundary data we conclude that

with a suitable function . By symmetry and so where is a suitable constant. Finally, the growth condition leads to , , and . ∎

Lemma 6 (Estimates near the boundary).

Suppose that and that is a bounded -smooth domain. Then for each there exists a radius and a constant such that for all one has

(12)
Proof.

We assume by contradiction that there exist such that

(13)

In particular we have , , . Without loss of generality we may assume that and that the first unit vector is the exterior unit normal to at . After possibly passing to a subsequence it is enough to consider one of the following two cases.


First case: .
This proof is as above for Lemma 4; only proving the required uniform bounds for the is slightly more involved. The arguments are sketched below in the second case. Thanks to Lemma 5 the convergence proof of [10, Lemma 7] can be extended to . One should observe that also the symmetry carries over to the limit.


Second case: .
Observe that in this case

and

The assumption gives that

(14)

In this case we rescale differently, however, denotes again the closest boundary point to . We introduce the rescaled biharmonic Green functions

for

Since , the exterior unit normal at converges to the first unit vector and so we conclude that

For

the assumption (14) transforms into

(15)

where . Since are bounded and their boundary distances are uniformly bounded from below by we find after passing to a further subsequence that . We claim that we have local uniform convergence in (including the diagonal) of to , since we are in dimension . To see this we observe first that Krasovskiĭ-type estimates (see [15, 16] and also [9, Theorem 4.20]) yield at a first instance useful information only for the third derivatives. We have

Making use of

and of

we obtain upon integration that

and further that

Now, one may proceed further as in [10, Lemma 7]. So, (15) yields , while Boggio’s formula shows that (even if ) since are interior points. ∎

Lemma 7 (Estimates in the interior).

Suppose that and that is a bounded -smooth domain. Then for each there exists a radius and a constant such that for all one has

(16)
Proof.

Since , we have that (see [17, p. 115]): since is continuous, there exists such that and for all . This yields (16) since is bounded. ∎

5. Proof of the main estimate for

Combining Lemmas 6 and 7 and applying a compactness argument to

we find:

Corollary 8.

Suppose that and that is a bounded -smooth domain. Then there exists a radius and a constant such that for all and for all one has

(17)

This step provides in particular a different and simpler proof for the local positivity statement from [9, Theorem 6.15] which was proved first by Dall’Acqua, Meister, and Sweers [4].

Proof of Theorem 1: If we take from [5], see also [9], that so that

Since we end up with

and positive constants in this case. The proof of Theorem 1 is now complete also for using Corollary 8.

6. Proof of Theorem 2

This theorem was proved in [10] when and in [9, Theorem 6.15] when . The latter proof is quite involved and based on the extensive use of conformal maps and explicit Green functions in certain limaçons. We provide here an alternate proof which uses the same techniques for as for .

Case I: . For this situation we have

(18)

Then there is such that we find for .

Case II: . Now we have

(19)

Since is bounded on one finds for the existence of such that for . For dimension the argument is more subtle. We fix as in Proposition 3. Taking for but sufficiently small it follows that . It remains to consider . Assume first that and . Then and we are in the situation just considered. So we are left with and . Then we may apply Theorem 1 of [12], see also Proposition 3, to find that for it follows that . For dimension the result follows directly from Corollary 8.


References

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