POSITIVITY ISSUES OF BIHARMONIC GREEN’S FUNCTIONS UNDER DIRICHLET BOUNDARY CONDITIONS

HANS-CHRISTOPH GRUNAU AND FRÉDÉRIC ROBERT

Dedicated to Prof. Wolf von Wahl on the occasion of his 65th birthday

Abstract. In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems.

It is shown that, on the other hand, for bounded smooth domains Ω ⊂ ℝ^{n}, the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n ≥ 3.

Moreover, the biharmonic Green’s function in balls B ⊂ ℝ^{n} under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n ≥ 3.

1. Introduction

Although simple examples show that strong maximum principles as satisfied e.g. by harmonic functions cannot hold true for solutions of higher order elliptic equations, it is reasonable to ask whether higher order boundary value problems may possibly enjoy a positivity preserving property. To be specific, we consider the clamped plate boundary value problem:

(1.1)  \{ \begin{matrix}Δ^{2}u = f in Ω, \\ u_{|_{∂Ω}} = \frac{∂}{∂ν} u_{|_{∂Ω}} = 0. \end{matrix}

Here Ω ⊂ ℝ^{n} is a bounded smooth domain with exterior unit normal ν at ∂Ω, and f is a sufficiently smooth datum. If n = 2, the unknown u models the vertical deflection of a horizontally clamped thin elastic plate from the horizontal equilibrium shape under the vertical load f. The boundary conditions u_{|_{∂Ω}} = \frac{∂}{∂ν} u_{|_{∂Ω}} = 0 are called Dirichlet boundary conditions and are natural in mechanics to model the horizontal clamping: More precisely, the condition u_{|_{∂Ω}} = 0 models the location of the clamping and the condition \frac{∂}{∂ν} u_{|_{∂Ω}} = 0 models the fact that the plate is clamped into some matter and is not able to rotate freely. We refer to the memoir written by Hadamard [16] for further considerations on this question. Throughout the present paper, always these boundary conditions will be considered.

We shall discuss comparison principles or positivity preserving properties for the biharmonic operator. We say that the clamped plate boundary value problem enjoys a positivity preserving property in Ω if the following assertion holds:

for all u ∈ C^{4}(\overline{Ω}) and f ∈ C^{0}(\overline{Ω}) satisfying (1.1), one has that \{ f ≥ 0 ⇒ u ≥ 0 \} .

This definition is the natural extension of the “positivity preserving property” for the harmonic operator, i.e. a comparison principle for a second-order operator. While the “positivity preserving property” is well understood for second-order operators, it is much more involved for fourth-order ones.

The positivity preserving property is closely related to the sign of the Green’s function. More precisely, let H_{Ω} = H_{Ω,Δ^{2}} be the singular Green’s function for the operator Δ^{2} in Ω under Dirichlet boundary conditions. Then, for any reasonable datum f : Ω → ℝ the solution u : \overline{Ω} → ℝ of the clamped plate boundary value problem (1.1) is given by

u(x) = ∫_{Ω} H_{Ω}(x, y)f(y) dy.

In particular, the clamped plate boundary value problem enjoys a positivity preserving property in Ω iff H_{Ω}(x, y) ≥ 0 for all x, y ∈ Ω, x ≠ y.

Date: August 18th 2008.

It is important to remark that positivity issues are related to the specific kind of prescribed boundary conditions. More precisely, if one chooses Navier boundary condition (that is u = Δu = 0 on ∂Ω), then a twofold application of the second order comparison principle immediately yields a positivity preserving property. This simple situation is misleading in several respects: As we will explain below, counterexamples show that the situation is much more intricate for other boundary conditions. For Dirichlet boundary conditions which we consider here, this iterative trick fails completely. Moreover, even under Navier conditions, there is in general no positivity preserving property for perturbations of the biharmonic operator, see [27, 19], cf. also the general approach in [6].

The first example of a positive Green’s function was given by Boggio [4] by means of a beautiful explicit formula for balls in ℝ^{n}, even for the Dirichlet problem for polyharmonic operators. Boggio [3] (1901) and Hadamard [16] (1908) conjectured that in arbitrary reasonable (two dimensional) domains Ω, the positivity preserving property should hold true. In 1909 Hadamard [17] already knew, that the positivity conjecture is false in annuli with small inner radius. However, there was still some hope to prove positivity for convex domains.

Starting about 40 years later, numerous counterexamples disproved the Boggio-Hadamard conjecture, see e.g. [9, 11, 28]. In particular, Coffman and Duffin [5] proved that in any two dimensional domain containing a right angle, the positivity preserving property does not hold. Even for smooth convex domains, the issue is quite intricate: Garabedian [11] (see also Shapiro-Tegmark [28]) proved that for long thin ellipses, the prositivity preserving property does not hold true. On the other hand, according to [13], one has positivity in ellipses, which are close enough to a ball. For a more extensive survey and further references we also refer to [13].

Therefore, the positivity preserving property does not hold true in general, even for arbitrarily smooth uniformly convex domains. Hence, it is important to understand the lack of “positivity preserving property” with the help of the Green’s function. One may ask the following questions:

(1) Is positivity preserving in any bounded smooth domain possibly “almost true” in the sense that the negative part H_{Ω}^{−}(x, y) := min{H_{Ω}(x, y), 0} of the biharmonic Green’s function under Dirichlet boundary conditions is “small relatively” to the singular positive part H_{Ω}^{+}(x, y) := max{H_{Ω}(x, y), 0}?

(2) Are there are at least families of domains, different from balls, where the biharmonic Green’s functions under Dirichlet boundary conditions are positive?

Question (1) is motivated by reactions of physicists and engineers on the mathematical results concerning positivity preserving and sign change. They may be summarised as follows: “For a clamped plate without corners, we do not expect downwards deflections if the force is pushing upwards. If such a phenomenon can be mathematically observed we think that perhaps the model is not perfectly suitable or that negativity is so small that it cannot be observed in reality.” Also numerical experiments give support to the second hypothesis, which is subject of our first main result.

The general behaviour of the Green’s functions is modeled on the behaviour of the singular fundamental solution on ℝ^{n}. On the whole space, we have that (letting e_{n} be the n-dimensional volume of B_{1}(0) ⊂ ℝ^{n})

H_{ℝ^{n}}(x, y) = \{ \begin{matrix}\frac{1}{2(n − 4)(n − 2)ne_{n}} |x − y|^{4−n}, & when n ≥ 5, \\ \frac{1}{16e_{4}} log \frac{1}{|x − y|}, & when n = 4, \\ −\frac{1}{8π} |x − y|, & when n = 3, \end{matrix}

for all x, y ∈ ℝ^{n}, x ≠ y. If n ≥ 5, this fundamental solution can even be interpreted as a Green’s function in ℝ^{n}, where the Dirichlet boundary conditions at infinity are understood as a suitable decay. In the general framework of a bounded domain of ℝ^{n}, Krasovskiĭ [20, 21] proved that there exists a constant C(Ω) such that

|H_{Ω}(x, y)| ≤ C(Ω) \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + | log |x − y| |) , & if n = 4, \\ 1, & if n = 3; \end{matrix}

for all x, y ∈ Ω, x ≠ y. These estimates give a uniform bound for the singular behaviour for Green’s functions but do not consider their boundary behaviour. The latter was done by Dall’Acqua and Sweers [8] by means of integrating Krasovskiĭ’s estimates for H and its derivatives.:

(1.2)  |H_{Ω}(x, y)| ≤ \{ \begin{matrix}C · |x − y|^{4−n} min \{1, \frac{d(x)^{2}d(y)^{2}}{|x − y|^{4}} \} , & if n > 4, \\ C · log (1 + \frac{d(x)^{2}d(y)^{2}}{|x − y|^{4}} ) , & if n = 4, \\ C · d(x)^{2−n/2}d(y)^{2−n/2} min \{1, \frac{d(x)^{n/2}d(y)^{n/2}}{|x − y|^{n}} \} , & if n = 2, 3. \end{matrix}

Here, d(x) = d(x, ∂Ω) and C = C(Ω) > 0 denotes a constant.

As far as the positive part H_{Ω}^{+} is concerned, these estimates cannot be improved, see e.g. [14]. However, they do not distinguish between the positive and the negative part of the Green’s function. A distinction between H_{Ω}^{+} and H_{Ω}^{−} and showing that pairs of points of negativity cannot approach each other is the subject of our first main result:

Theorem 1. Let Ω ⊂ ℝ^{n} (n ≥ 3) be a bounded C^{4,α}-smooth domain. We denote by H_{Ω} the biharmonic Green’s function in Ω under Dirichlet boundary conditions. Then, there exists a constant δ = δ(Ω) > 0 such that for any two points x, y ∈ Ω, x ≠ y,

H_{Ω}(x, y) ≤ 0 implies that |x − y| ≥ δ.

Consequently, there exists a constant C = C(Ω) > 0 such that for any two points x, y ∈ Ω, x ≠ y, we have that

(1.3)  H_{Ω}(x, y) ≥ −C(Ω),

i.e. the negative part H^{−} of the Green’s function is bounded. Moreover, if Ω is smooth enough for (1.2) to hold true, the estimate (1.3) from below can be refined:

(1.4)  H_{Ω}(x, y) ≥ −C(Ω) d(x)^{2} d(y)^{2}.

In other words: Around the pole, biharmonic Green’s functions are always positive, if n ≥ 3. And this behaviour is uniform, even if the pole approaches the boundary.

The proof of Theorem 1 indicates that one may not expect the full result to hold true also for n = 2. The bound (1.4), however, was proved for the case n = 2 and sufficiently smooth domains by Dall’Acqua, Meister and Sweers [7]. Even for n = 2, 3, where the Green’s function is bounded, (1.4) is a strong statement because in the case, where x or y is closer to the boundary than they are to each other, (1.2) would only give H_{Ω} ≥ −C \frac{d(x)^{2}d(y)^{2}}{|x−y|^{n}} . In this sense, (1.4) gains a factor of order |x − y|^{n}.

Another consequence of Theorem 1 is that one has a strong control of the negative part of solutions u of the clamped plate boundary value problem (1.1) with nonnegative datum f irrespective of the space dimension:

f ≥ 0 ⇒ ∥u^{−}∥_{L^{∞}(Ω)} ≤ C(Ω)∥f∥_{L^{1}(Ω)}.

This estimate should be compared with the estimate for the full function: It follows from general elliptic theory [1] that for all p > \frac{n}{4}, there exists C(p, Ω) > 0 such that for solutions u of the clamped plate boundary value problem (1.1) with datum f, we have that

∥u∥_{L^{∞}(Ω)} ≤ C(p, Ω)∥f∥_{L^{p}(Ω)}.

Consequently, on the one hand, there might be a negative part for u, even if f ≥ 0. But on the other hand, this negative part enjoys a nice strong control by a very weak norm. Therefore, although one has a lack of positivity in general, in this sense it is “small”.

Let us now turn to Question (2): Are there at least families of domains, different from balls, where the biharmonic Green’s functions under Dirichlet boundary conditions are positive? For two dimensional domains, this question was addressed in [13]. There, it was shown that in domains Ω ⊂ ℝ^{2} being sufficiently close in C^{4}-sense to the (unit) disk B ⊂ ℝ^{2}, the biharmonic Green’s function (under Dirichlet boundary conditions) is positive. Recently, Sassone [24] could relax the assumption on the domains to be close to B in C^{2,α}-sense. The authors could take advantage of conformal maps and the Riemann mapping theorem, pulling back the clamped plate boundary value problem to Dirichlet problems in the unit disk with the biharmonic operator as principal part and only with (small) lower order perturbations. The latter were treated in B ⊂ ℝ^{n} (n arbitrary) in [14]. The methods of [13], however, do not carry over to higher dimensions due to a lack of sufficiently many conformal maps. So, the question, whether the positivity of the biharmonic Green’s function in the unit ball B ⊂ ℝ^{n} is stable under domain perturbations, was left open.

This question is addressed in our next result. Assuming n > 2, we show that in domains Ω ⊂ ℝ^{n}, which are sufficiently close to the unit ball in a suitable C^{4,α}-sense, the biharmonic Green’s function under Dirichlet boundary conditions is indeed positive. More precisely, we prove the following theorem, where Id denotes the identity map:

Theorem 2. Let B be a unit ball of ℝ^{n}, n ≥ 3. Then, there exists ε_{0} = ε_{0}(n) > 0 such that the following holds true:

We assume that Ω ⊂ ℝ^{n} is a C^{4,α}-smooth domain which is ε_{0}-close to the ball B in the C^{4,α}-sense, i.e.: There exists a surjective C^{4,α}-map ψ : \overline{B} → \overline{Ω} such that ∥Id − ψ∥_{C^{4,α}(\overline{B})} ≤ ε_{0}.

Then, the Green’s function H_{Ω} for Δ^{2} in Ω under Dirichlet boundary conditions is strictly positive:

∀x, y ∈ Ω, x ≠ y : H_{Ω}(x, y) > 0.

Assuming ε_{0} small enough, this notion of closeness implies that there is a fixed neighbourhood U of \overline{B}, C^{4,α}-smooth injective extensions ψ : \overline{U} → ℝ^{n}, ∥Id − ψ∥_{C^{4,α}(\overline{U})} = O(ε) and C^{4,α}-smooth inverse maps

ϕ = ψ^{−1} : ψ(\overline{U}) → \overline{U} such that ψ (\overline{B}) = \overline{Ω}, ψ (B) = Ω.

For n = 2, a direct and explicit proof based on perturbation series, Green’s function estimates and conformal maps was given in [13, 14]. This means that there, in principle, ε_{0} may be calculated explicitly. Moreover, in the case n = 2, closeness has to be assumed only in a weaker norm, see [24].

Here, the proof is somehow more indirect since a number of proofs by contradiction is involved so that it will be impossible to calculate ε_{0} from our proofs. Furthermore, we have to make extensive use of general elliptic theory as provided by Agmon, Douglis and Nirenberg [1]. We emphasize that Theorem 2 is by no means just a continuous dependence on data result, since the involved Green’s “functions” are not simply functions but families of functions depending on the position of the singularity.

The problem consists in gaining uniformity with respect to the position of the singularity: When the singularity is in the interior, it is possible to use the local positivity results of Grunau-Sweers [15]. But when the singularity approaches the boundary, the situation becomes more intricate and we develop a new technique for this problem. These remarks apply also to Theorem 1 which is proved by means of the same methods.

It remains as an interesting question to find out an optimal notion of closeness for a result like Theorem 2 to hold true. One might expect that like in the two-dimensional case (see[24]) C^{2,α}-closeness could suffice: Does the boundary curvature govern the positivity behaviour of biharmonic Green’s functions? However, such a result would require new ideas and techniques; its possible proof would certainly be much more technical than ours. Sassone’s approach is strictly two-dimensional because of his use of conformal maps.

Our methods and techniques are general and our results can be extended to more general fourth order “positive definite self adjoint” elliptic operators under Dirichlet boundary conditions, where the principal part is a square of second order elliptic operators, and also to similar elliptic operators of higher order 2m in dimensions n ≥ 2m−1 with reasonable boundary conditions of the type discussed in Agmon-Douglis-Nirenberg [1].

2. A more general result

In order to prove Theorem 2, below in Theorem 3 we describe the possible situations how transition from positivity to sign change may occur within a smooth family of domains. It is then easy to see that none of these situations occurs in the (unit) ball in ℝ^{n}, n > 2. Moreover, a special case of Theorem 3 will directly yield the proof of Theorem 1.

To provide a more flexible result in Theorem 3, we will also include lower order perturbations. The formulation is somehow technical and requires in particular the notion of smooth domain perturbations, which we make precise in the following definition.

Definition 1. Let Ω, (Ω_{k})_{k∈ℕ} be domains of ℝ^{n}. We say that (Ω_{k})_{k∈ℕ} is a C^{4,α}-smooth perturbation of the bounded C^{4,α}-smooth domain Ω, and we write

lim_{k→+∞} Ω_{k} = Ω

if the following facts are satisfied:

(i) There exist N ∈ ℕ, p_{1}, …, p_{N} ∈ ∂Ω, δ > 0 and open subsets ω ⊂⊂ Ω, ω ⊂⊂ ω_{0} ⊂⊂ Ω_{k} such that

Ω ⊂ ω ∪ ⋃_{i=1}^{N} B_{δ}(p_{i}); Ω_{k} ⊂ ω ∪ ⋃_{i=1}^{N} B_{δ}(p_{i});

(ii) for any i ∈ {1, …, N}, there exists an open subset of U_{i} ⊂ ℝ^{n} such that 0 ∈ U_{i}, and a C^{4,α}-smooth diffeomorphism Φ_{i} : U_{i} → B_{2δ}(p_{i}) such that Φ_{i}(0) = p_{i} and

Φ_{i}(U_{i} ∩ {x_{1} < 0}) = Φ_{i}(U_{i}) ∩ Ω , Φ_{i}(U_{i} ∩ {x_{1} = 0}) = Φ_{i}(U_{i}) ∩ ∂Ω;

(iii) for any i ∈ {1, …, N} and k ∈ ℕ, there exists Φ_{k,i} : U_{i} → B_{2δ}(p_{i}) such that Φ_{k,i}(U_{i}) is an open subset, B_{3δ/2}(p_{i}) ⊂ Φ_{k,i}(U_{i}) and Φ_{k,i} : U_{i} → Φ_{k,i}(U_{i}) is a C^{4,α}-smooth diffeomorphism and

Φ_{k,i}(U_{i} ∩ {x_{1} < 0}) = Φ_{k,i}(U_{i}) ∩ Ω_{k} , Φ_{k,i}(U_{i} ∩ {x_{1} = 0}) = Φ_{k,i}(U_{i}) ∩ ∂Ω_{k};

(iv) for any i ∈ {1, …, N}, lim_{k→+∞} Φ_{k,i} = Φ_{i} in C_{loc}^{4,α}(U_{i}).

This definition implies that we have a well defined smooth exterior normal vector field so that Ω, Ω_{k}, ∂Ω and ∂Ω_{k} carry a canonical orientation. In what follows, the local charts will be chosen such that this orientation is observed, i.e. such that Jac Φ_{i} ∘ Φ_{j}^{−1} > 0, Jac Φ_{k,i} ∘ Φ_{k,j}^{−1} > 0.

This definition covers in particular the following more special situation of smooth domain perturbation, which we make use of in proving Theorem 2: Let a sequence of mappings (ψ_{k})_{k∈ℕ} be such that there exists an open subset of U ⊂ ℝ^{n} and ψ_{k} : U → ℝ^{n} for all k ∈ ℕ. We assume that lim_{k→+∞} ψ_{k} = Id in C_{loc}^{4,α}(U). Let Ω ⊂⊂ U be a C^{4,α}-smooth bounded subset of ℝ^{n} and let Ω_{k} := ψ_{k}(Ω) for all k ∈ ℕ. Then the sequence (Ω_{k})_{k∈ℕ} is a smooth perturbation of Ω.

Employing the notion of smooth domain perturbation we are now able to formulate our key result (where Theorems 1 and 2 are a consequence of as it is explained at the end of Section 6):

Theorem 3. Let n ≥ 3, and (Ω_{k})_{k∈ℕ} be a C^{4,α}-smooth perturbation of the bounded C^{4,α}-smooth domain Ω in the sense of Definition 1. We consider a sequence (a_{k})_{k∈ℕ} ∈ C^{0,α}(U_{0}), where Ω ⊂⊂ U_{0} and assume that there exists a_{∞} ∈ C^{0,α}(U_{0}) such that

lim_{k→+∞} a_{k} = a_{∞} in C_{loc}^{0,α}(U_{0}).

We assume further that there exists λ > 0 such that

(2.1)  ∫_{Ω_{k}} ((Δφ)^{2} + a_{k}φ^{2}) dx ≥ λ ∫_{Ω_{k}} φ^{2} dx

for all φ ∈ C_{c}^{∞}(Ω_{k}) and all k ∈ ℕ. Let G_{k} be the Green’s function of Δ^{2} + a_{k} on Ω_{k}, and G be the Green’s function of Δ^{2} + a_{∞} on Ω, all with Dirichlet boundary conditions.

Finally, we suppose that there exist two sequences (x_{k})_{k∈ℕ}, (y_{k})_{k∈ℕ} such that x_{k}, y_{k} ∈ Ω_{k} and

G_{k}(x_{k}, y_{k}) = 0 for all k ∈ ℕ.

Up to a subsequence, let x_{∞} := lim_{k→+∞} x_{k} and y_{∞} := lim_{k→+∞} y_{k}. Then x_{∞}, y_{∞} ∈ \overline{Ω}, x_{∞} ≠ y_{∞} and we are in one of the following situations:

(i) if x_{∞}, y_{∞} ∈ Ω, then G(x_{∞}, y_{∞}) = 0; (ii) if x_{∞} ∈ Ω and y_{∞} ∈ ∂Ω, then Δ_{y}G(x_{∞}, y_{∞}) = 0; (iii) if x_{∞} ∈ ∂Ω and y_{∞} ∈ Ω, then Δ_{x}G(x_{∞}, y_{∞}) = 0; (iv) if x_{∞}, y_{∞} ∈ ∂Ω, then Δ_{x}Δ_{y}G(x_{∞}, y_{∞}) = 0.

In the above statement, Δ_{x}G denotes the Laplacian with respect to the first variables, and Δ_{y}G denotes the Laplacian with respect to the second variables. The uniform coercivity assumption (2.1) is e.g. implied by a sign condition like a_{k} ≥ 0 and uniform smoothness of the domains to have a uniform Poincaré-Friedrichs inequality.

More general lower order “self adjoint” perturbations of the biharmonic operator may be covered by precisely the same techniques. However, here we prefer to stick to a relatively simple situation in order to avoid too many technical details.

In the one dimensional context (clamped bars), related and quite concrete results were obtained by Schröder [25, 26, 27].

In order to gain a better feeling for the statement of Theorem 3 one should think of deforming the ball, where we know that positivity preserving holds true, smoothly into a domain where the biharmonic Green’s function is sign changing (e.g. a long thin ellipsoid). There is a “last” domain where one still has a nonnegative Green’s function. Our result describes the possible degeneracies of this positivity via which sign change occurs beyond this “last positivity-domain”. The key statement is that x_{∞} ≠ y_{∞} so that it is impossible that negativity appears through the singularity: Around the singularity, our Green’s functions are always positive. The most difficult part is to prove this also arbitrarily close to the boundary.

Alternatively one may think of the Green’s function for Δ^{2} + λ in a ball for λ → ∞ where again, initially one has positivity while sign changes occur for λ → ∞ (see e.g. [2, 6]). We think that most likely the transition from positivity to sign change will occur via alternative (iv) of Theorem 3.

Throughout the paper we assume that

n ≥ 3.

A first essential step in proving Theorem 3 consists in providing uniform bounds (in k) for the Green’s functions like

(2.2)  |G_{k}(x, y)| ≤ C · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4, \\ 1, & if n = 3. \end{matrix}

Moreover, if n = 3, 4, the somehow irregular estimates for G_{k} require to focus first on the gradients, where estimates like

(2.3)  |∇G_{k}(x, y)| ≤ C · \{ \begin{matrix}|x − y|^{−1} , & if n = 4, \\ 1, & if n = 3, \end{matrix}

are available, which are compatible with the scaling arguments performed below. In this respect, the proof is more difficult in dimensions n = 3 and in particular n = 4. Estimates (2.2) and (2.3) are due to Krasovskiĭ [21], provided the family (Ω_{k}) is assumed to be uniformly C^{11}-smooth. This assumption is due to the great generality of the boundary value problems considered by Krasovskiĭ. We prove in Theorem 4 that in our special situation, (2.2) and (2.3) hold true under our uniform C^{4,α}-smoothness assumptions. Preliminary properties of the Green’s functions are shown in Section 3, while Section 5 is devoted to convergence properties of families of Green’s functions in (Ω_{k})_{k∈ℕ}. The proofs of Theorems 3 and, as a consequence, of Theorems 1 and 2 are finally given in Section 6.

Notation. Straightening the boundary requires to work in ℝ_{−}^{n} := {x ∈ ℝ^{n} : x_{1} < 0}, where we write ℝ^{n} ∋ x = (x_{1}, x̄). e_{n} denotes the n-dimensional volume of B_{1}(0) ⊂ ℝ^{n}. C_{c}^{∞}(Ω) denotes the space of arbitrarily smooth functions with compact support in Ω and D′(Ω) its dual, i.e. the space of distributions on Ω.

3. The Green’s function G for the perturbed biharmonic operator

In the first part of this section, we consider a fixed operator Δ^{2} + a in a fixed smooth domain and construct and investigate the corresponding Green’s function. The construction is standard, but we recall it since we need a precise control of the constants in particular in view of the uniform estimates in Section 4.

Proposition 3.1. Let Ω ⊂ ℝ^{n} be a bounded C^{4,α}-smooth domain and a ∈ C^{0,α} (\overline{Ω}). We assume that Δ^{2} +a is coercive. Then, for every x ∈ Ω, there exists a unique Green’s function G_{x} ∈ L^{1}(Ω) ∩ C^{4,α} (\overline{Ω} \ \{x\}) such that G_{x}|_{∂Ω} = \frac{∂G_{x}}{∂ν} |_{∂Ω} = 0 and that for all φ ∈ C^{4} (\overline{Ω}) with φ|_{∂Ω} = \frac{∂φ}{∂ν}|_{∂Ω} = 0 one has the following representation formula:

(3.1)  φ(x) = ∫_{Ω} G_{x}(y) (Δ^{2}φ(y) + a(y)φ(y)) dy.

If R > 0 is such that Ω ⊂ B_{R}(0) and λ > 0 such that ∀φ ∈ W_{0}^{2,2}(Ω) : ∫_{Ω} ((Δφ)^{2} + aφ^{2}) dy ≥ λ ∫_{Ω} φ^{2} dy then, the following estimate for the Green’s function holds true:

(3.2) |G_{x}(y)| ≤ C(λ, R, n, ∥a∥_{C^{0,α}}, Ω) · \{ \begin{matrix}(|x − y|^{4−n} + max\{d(x, ∂Ω), d(y, ∂Ω)\}^{4−n}) , & if n > 4, \\ 1 + |log |x − y|| + |log (max\{d(x, ∂Ω), d(y, ∂Ω)\})| , & if n = 4, \\ 1, & if n = 3. \end{matrix}

If n = 3, 4, we further prove the following gradient estimates:

(3.3) |∇_{(x,y)}G_{x}(y)| ≤ C(λ, R, n, ∥a∥_{C^{0,α}}, Ω) · \{ \begin{matrix}(|x − y|^{−1} + max\{d(x, ∂Ω), d(y, ∂Ω)\}^{−1}) , & if n = 4, \\ 1, & if n = 3. \end{matrix}

The dependence of the constants C on Ω is explicit via the C^{4,α}-properties of ∂Ω.

Proof. We first prove extensively the generic case n > 4. At the end we comment on the changes and additional arguments which have to be made, if n = 4 or n = 3.

Case n > 4. We introduce the fundamental solution Γ_{0} of the biharmonic operator

(3.4)  Γ_{0}(x, y) := \frac{1}{2(n − 4)(n − 2)ne_{n}} |x − y|^{4−n}

so that Γ_{0} ∈ C^{∞} (\overline{Ω} × \overline{Ω}) \ \{(x, y) : x = y\}. We define recursively for j ≥ 0

Γ_{j+1}(x, y) := − ∫_{Ω} Γ_{j}(x, z)a(z)Γ_{0}(z, y) dz

and have that Γ_{j} ∈ C^{4,α} (\overline{Ω} × \overline{Ω} \ \{(x, y) : x = y\}) is well defined and, according to a Lemma of Giraud [12], that for j ≥ 1

(3.5)  |Γ_{j}(x, y)| ≤ \{ \begin{matrix}C_{j}|x − y|^{4(j+1)−n}, & if (j + 1) < \frac{n}{4}, \\ C_{j} (1 + |log |x − y||) , & if (j + 1) = \frac{n}{4}, \\ C_{j}, & if (j + 1) > \frac{n}{4}. \end{matrix}

Here, C_{j} = C_{j} (n, R, ∥a∥_{∞}), where R > 0 is chosen such that Ω ⊂ B_{R}(0). We fix some ℓ > \frac{n}{4}, x ∈ Ω and for u_{x} ∈ C^{4,α} (\overline{Ω}) to be suitably determined below, we put

(3.6)  G_{x}(y) := Γ_{0}(x, y) + ∑_{j=1}^{ℓ} Γ_{j}(x, y) + u_{x}(y).

One should observe that ∑_{j=0}^{∞} Γ_{j} is the Neumann-series for the fundamental solution for the perturbed differential operator. We have that G_{x} ∈ C^{4,α} (\overline{Ω} \ \{x\}). In order that G_{x} becomes indeed a Green’s function for the Dirichlet problem for Δ^{2} + a, i.e. that indeed formula (3.1) is satisfied, we need u_{x} to be a solution of the following Dirichlet problem

(3.7)  \{ \begin{matrix}Δ^{2}u_{x}(y) + a(y)u_{x}(y) = −a(y)Γ_{ℓ}(x, y) & in Ω \\ u_{x}(y) = −Γ_{0}(x, y) − ∑_{j=1}^{ℓ} Γ_{j}(x, y) & for y ∈ ∂Ω, \\ \frac{∂}{∂ν} u_{x}(y) = −\frac{∂}{∂ν_{y}} Γ_{0}(x, y) − ∑_{j=1}^{ℓ} \frac{∂}{∂ν_{y}} Γ_{j}(x, y) & for y ∈ ∂Ω. \end{matrix}

Since ℓ > \frac{n}{4}, the right hand side −a · Γ_{ℓ}(x, .) is Hölder continuous with Hölder norm bounded by a constant C(n, R, ∥a∥_{C^{0,α}}). The C^{1,α}-norm of the datum for u_{x}|_{∂Ω} and the C^{0,α}-norm of the datum for \frac{∂}{∂ν} u_{x}|_{∂Ω} are bounded by C(n, R, ∂Ω)d(x, ∂Ω)^{3−n−α}. The dependence of the constant C on ∂Ω is in principle constructive and explicit via its curvature properties and their derivatives. According to C^{1,α}–estimates for boundary value problems in variational form like (3.7) – see [1, Thm. 9.3] – we see that

(3.8)  ∥u_{x}∥_{C^{1,α}(\overline{Ω})} ≤ C(n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω)d(x, ∂Ω)^{3−n−α}.

One should observe that the differential operators are uniformly coercive, so that no u_{x}-terms need to appear on the right-hand-side.

As long as d(y, ∂Ω) ≤ d(x, ∂Ω), (3.8) shows that

|u_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω)d(x, ∂Ω)^{4−n}

and hence

(3.9)  |G_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{4−n} + d(x, ∂Ω)^{4−n}) .

If d(y, ∂Ω) > d(x, ∂Ω) we conclude from (3.9) by exploiting the symmetry of the Green’s function:

(3.10)  |G_{x}(y)| = |G_{y}(x)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{4−n} + d(y, ∂Ω)^{4−n}) .

Combining (3.9) and (3.10) yields (3.2) for n > 4.

Case n = 4. Here the fundamental solution we start with is

(3.11)  Γ_{0}(x, y) := −\frac{1}{16e_{4}} log |x − y|.

We proceed with the iterated kernels Γ_{j}. In view of the mild singularity of Γ_{0}, however, it is sufficient to choose ℓ = 1. As above we find that

(3.12)  ∥u_{x}∥_{C^{1,α}(\overline{Ω})} ≤ C(n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω)d(x, ∂Ω)^{−1−α}.

As long as d(y, ∂Ω) ≤ d(x, ∂Ω), (3.12) shows that

(3.13)  |∇_{y}G_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{−1} + d(x, ∂Ω)^{−1}) .

In order to exploit the symmetry of G_{x}(y) we need a similar estimate also for |∇_{x}G_{x}(y)|. To this end one has to differentiate (3.7) with respect to the parameter (!) x and obtains as before that for d(y, ∂Ω) ≤ d(x, ∂Ω)

(3.14)  |∇_{x}G_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{−1} + d(x, ∂Ω)^{−1}) .

By symmetry G_{x}(y) = G_{y}(x), and (3.14) shows that for d(x, ∂Ω) ≤ d(y, ∂Ω), one has

(3.15)  |∇_{y}G_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{−1} + d(y, ∂Ω)^{−1})

while (3.13) yields

(3.16)  |∇_{x}G_{x}(y)| ≤ C(C_{0}, n, R, λ, ∥a∥_{C^{0,α}}, ∂Ω) (|x − y|^{−1} + d(y, ∂Ω)^{−1}) .

Combining (3.13)-(3.16) proves (3.3) and hence (3.2) in the case n = 4.

Case n = 3. Here, we simply work with the bounded Lipschitz continuous fundamental solution

(3.17)  Γ_{0}(x, y) := −\frac{1}{8π} |x − y|

so that no iterative procedure is needed and we may directly work with ℓ = 0. One comes up with

(3.18)  ∥u_{x}∥_{C^{1,α}(\overline{Ω})} ≤ C(R, n, λ, ∥a∥_{C^{0,α}}, ∂Ω)d(x, ∂Ω)^{−α}.

Proceeding as for n = 4 yields (3.3) and hence (3.2) also in the case n = 3. □

Let us now show that assuming certain uniform estimates on the Green’s functions H_{k} for the biharmonic operators on a family (Ω_{k}) of domains according to Definition 1 implies similar uniform estimates for the Green’s functions of the perturbed biharmonic operators Δ^{2} + a_{k} on Ω_{k}:

Proposition 3.2. Let n ≥ 4 and (Ω_{k})_{k∈ℕ} be a C^{4,α}-smooth perturbation of the bounded C^{4,α}-smooth domain Ω according to Definition 1 and R > 0 such that Ω_{k} ⊂ B_{R}(0). Let H_{k} ∈ C^{4} (\overline{Ω_{k}} × \overline{Ω_{k}} \ \{(x, y) : x = y\}) denote the Green’s functions for Δ^{2} in Ω_{k} and assume that there exists a uniform constant C_{1} such that

(3.19)  ∀k ∀x, y ∈ Ω_{k}(x ≠ y) :  |H_{k}(x, y)| ≤ C_{1} · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4. \end{matrix}

Let a_{k} ∈ C^{0,α} (\overline{Ω_{k}}) and Λ > 0 such that ∀k : ∥a_{k}∥_{C^{0,α}(\overline{Ω_{k}})} ≤ Λ and let λ > 0 be such that

∀k ∀φ ∈ C_{c}^{∞}(Ω_{k}) : ∫_{Ω_{k}} ((Δφ)^{2} + a_{k}φ^{2}) dy ≥ λ ∫_{Ω_{k}} φ^{2} dy.

We denote by G_{k} the Green’s functions for Δ^{2}+a_{k} in Ω_{k}. Then, there exists a constant C_{2} = C_{2}(R, n, C_{1}, λ, Λ, Ω) such that one has the following estimate:

(3.20)  ∀x, y ∈ Ω_{k}, x ≠ y :  |G_{k}(x, y)| ≤ C_{2} · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4. \end{matrix}

Moreover, assuming

(3.21)  ∀x, y ∈ Ω_{k}, x ≠ y :  |∇_{(x,y)}H_{k}(x, y)| ≤ C_{1} |x − y|^{−1} ,  if n = 4,

in dimension n = 4 implies that

(3.22)  ∀x, y ∈ Ω_{k}, x ≠ y :  |∇_{(x,y)}G_{k}(x, y)| ≤ C_{2} |x − y|^{−1} ,  if n = 4.

The dependence on (Ω_{k})_{k} as regular perturbations of Ω is explicit via the geometric properties of ∂Ω. As long as these properties are uniformly satisfied, the same constant may be chosen.

The case n = 3 need not be covered here, since in this case, Proposition 3.1 already provides strong enough information for our purposes.

Proof. We proceed quite similarly as in the proof of Proposition 3.1, but now using the biharmonic Green’s functions H_{k} instead of Γ_{0}. That means that in Ω_{k}, we define inductively

Γ_{k,1}(x, y) := − ∫_{Ω_{k}} H_{k}(x, z)a_{k}(z)H_{k}(z, y) dz;

Γ_{k,j+1}(x, y) := − ∫_{Ω_{k}} Γ_{k,j}(x, z)a_{k}(z)H_{k}(z, y) dz.

Moreover, as above, we make the ansatz with u_{k,x} ∈ C^{4,α}(\overline{Ω_{k}})

(3.23)  G_{k}(x, y) := H_{k}(x, y) + ∑_{j=1}^{ℓ} Γ_{k,j}(x, y) + u_{k,x}(y).

We choose ℓ > \frac{n}{4} + 1 so that

(3.24)  |Γ_{k,ℓ}| ≤ C(R, n, Λ),  |∇Γ_{k,ℓ}| ≤ C(R, n, Λ),

while for the other Γ_{j}, we have in particular that

(3.25)  |Γ_{k,j}(x, y)| ≤ C(R, n, Λ) · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4, \end{matrix}

and assuming (3.21) that

(3.26)  ∀x, y ∈ Ω_{k}, x ≠ y :  |∇_{(x,y)}Γ_{k,j}(x, y)| ≤ C(R, n, Λ) · |x − y|^{−1} , if n = 4.

As before we see that G_{k} is indeed the Green’s function for the Dirichlet problem for Δ^{2} + a_{k} in Ω_{k}, iff the u_{k,x} solve the following boundary value problems:

(3.27)  \{ \begin{matrix}Δ^{2}u_{k,x}(y) + a_{k}(y)u_{k,x}(y) = −a_{k}(y)Γ_{k,ℓ}(x, y) & in Ω_{k} \\ u_{k,x}(y) = \frac{∂}{∂ν} u_{k,x} = 0 & for y ∈ ∂Ω_{k}. \end{matrix}

The right hand side is uniformly bounded, the operators are uniformly coercive. Hence, L^{p}-theory (see [1]) combined with Sobolev embedding theorems and differentiating (3.27) with respect to the parameter x yields

(3.28)  |u_{k,x}(y)| ≤ C(R, n, C_{1}, λ, Λ, (Ω_{k})_{k∈ℕ}),  |∇_{(x,y)}u_{k,x}(y)| ≤ C(R, n, C_{1}, λ, Λ, (Ω_{k})_{k∈ℕ}).

The dependence on (Ω_{k})_{k} is uniform in the sense described before the present proof. Inserting (3.24), (3.25), (3.28) and (3.26) into (3.23) proves the claim. □

Finally, we need a more precise statement concerning the smoothness of the Green’s functions simultaneously with respect to both variables.

Proposition 3.3. Under the assumptions of Proposition 3.2 we have in addition that

G_{k} ∈ C^{4,α} (\overline{Ω_{k}} × \overline{Ω_{k}} \ \{(x, y) : x ≠ y\}) .

Proof. We let i ∈ \{0, …, 3\} and p ∈ (n, n + 1) so that in particular 4 − i − \frac{n}{p} > 0. We let φ ∈ C_{c}^{∞}(Ω_{k}) and consider ψ ∈ C^{4,α}(\overline{Ω_{k}}) such that Δ^{2}ψ + a_{k}ψ = φ in Ω_{k} and ψ = ∂_{ν}ψ = 0 on ∂Ω_{k}. It follows from regularity theory (see [1]) and Sobolev’s embedding theorem that

∥ψ∥_{C^{i,β}(\overline{Ω_{k}})} ≤ C∥ψ∥_{W^{4,p}(Ω_{k})} ≤ C∥φ∥_{L^{p}(Ω_{k})}

with β ≤ 4 − i − \frac{n}{p}, β ∈ (0, 1). Here W^{4,p} denotes the Sobolev space of order 4 in differentiability and of order p in integrability. Since ψ(x) = ∫_{Ω_{k}} G_{k}(x, y)φ(y) dy, we get that ∇_{x}^{i}G_{k} makes sense and that

|∫_{Ω_{k}}(∇_{x}^{i}G_{k}(x, y) − ∇_{x}^{i}G_{k}(x′, y))φ(y) dy| ≤ C_{2}∥φ∥_{L^{p}(Ω_{k})}|x − x′|^{β}.

By duality, we then get that y ↦ ∇_{x}^{i}G_{k}(x, y) ∈ L^{q}(Ω_{k}) for all q ∈ (\frac{n+1}{n}, \frac{n}{n−1}) and that

∥∇_{x}^{i}G_{k}(x, ·) − ∇_{x}^{i}G_{k}(x′, ·)∥_{q} ≤ C(q)|x − x′|^{β} for all x, x′ ∈ Ω_{k}.

It follows from the equation satisfied by G_{k}(x, ·) that we have that Δ^{2}∇_{x}^{i}G_{k}(x, ·) + a∇_{x}^{i}G_{k}(x, ·) = 0 in D′(Ω_{k} \ {x}) and ∇_{x}^{i}G_{k}(x, ·) = 0, ∂_{ν}∇_{x}^{i}G_{k}(x, ·) = 0 on ∂Ω_{k}. It then follows from regularity theory that ∇_{x}^{i}G_{k}(x, ·) ∈ C^{4,α}(\overline{Ω_{k}} \ \{x\}). Moreover, for all δ > 0, there exists C(δ) > 0 such that

∥∇_{x}^{i}G_{k}(x, ·) − ∇_{x}^{i}G_{k}(x′, ·)∥_{C^{4,α}(\overline{Ω_{k}}\(B_{δ}(x)∪B_{δ}(x′)))} ≤ C(δ)|x − x′|^{β} for all x, x′ ∈ Ω_{k}.

This is valid for i ≤ 3; using the symmetry of the Green’s function, we have a similar result for i = 4 with respect to the C^{3,α}(\overline{Ω_{k}} \ (B_{δ}(x) ∪ B_{δ}(x′)))-norm. It then follows that all derivatives of order 4 are covered so that G_{k} ∈ C^{4,α}(\overline{Ω_{k}} × \overline{Ω_{k}} \ \{(x, x) : x ∈ \overline{Ω_{k}}\}). This proves the proposition. □

4. Uniform bounds for the Green’s functions

As before, we consider a family of bounded regular domains (Ω_{k})_{k∈ℕ} being a smooth perturbation of a fixed bounded regular domain Ω according to Definition 1. We focus on proving

|H_{k}(x, y)| ≤ C_{1} \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + | log |x − y| |) , & if n = 4, \\ 1, & if n = 3; \end{matrix}

|∇_{(x,y)}H_{k}(x, y)| ≤ C_{1} \{ \begin{matrix}|x − y|^{−1}, & if n = 4, \\ 1, & if n = 3; \end{matrix}

with the constant C_{1} = C_{1}(Ω) being uniform for the whole family (Ω_{k})_{k∈ℕ}. Originally, this type of estimates on the Green’s functions is due to Krasovskiĭ [21] (cf. also [20]) even for very general boundary value problems for even order elliptic operators. For the reader’s convenience, we include here an independant and shorter proof of these estimates.

Theorem 4. Let Ω be a bounded C^{4,α}-smooth domain of ℝ^{n}, n ≥ 3 and (Ω_{k})_{k∈ℕ} a C^{4,α}-smooth perturbation of Ω. We denote by H_{k} the Green’s functions for Δ^{2} in Ω_{k} under Dirichlet boundary conditions.

Then, there exists a constant C_{1} > 0 such that for all k and all x, y ∈ Ω_{k} with x ≠ y one has that

(4.1)  |H_{k}(x, y)| ≤ C_{1} · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4, \\ 1, & if n = 3. \end{matrix}

Moreover, for n = 3, 4 we prove that

(4.2)  ∀x, y ∈ Ω_{k}, x ≠ y :  |∇_{(x,y)}H_{k}(x, y)| ≤ C_{1} · \{ \begin{matrix}|x − y|^{−1} , & if n = 4, \\ 1, & if n = 3. \end{matrix}

Proof. If n = 3, the statement of Proposition 3.1 is already strong enough and nothing remains to be proved. We postpone the case n = 4 and start with proving the theorem in the generic case n > 4. We argue by contradiction and assume that there exist two sequences (x_{k})_{k∈ℕ}, (y_{k})_{k∈ℕ} with x_{k}, y_{k} ∈ Ω_{ℓ_{k}} such that x_{k} ≠ y_{k} for all k ∈ ℕ and such that

(4.3)  lim_{k→+∞} |x_{k} − y_{k}|^{n−4}|H_{ℓ_{k}}(x_{k}, y_{k})| = +∞.

It is enough to consider ℓ_{k} = k; other situations may be reduced to this by relabelling or are even more special. After possibly passing to a subsequence, it follows from (3.2) that there exists x_{∞} ∈ ∂Ω such that

(4.4)  lim_{k→+∞} x_{k} = x_{∞} and lim_{k→+∞} \frac{d(x_{k}, ∂Ω_{k})}{|x_{k} − y_{k}|} = 0.

We remark that the constant in (3.2) can be chosen uniformly for the family (Ω_{k})_{k∈ℕ}.

Lemma 4.1. Assume that n ≥ 4. For any q ∈ (\frac{n}{n−3}, \frac{n}{n−4}), there exists C(q) > 0 such that for all k and all x ∈ Ω_{k} we have

(4.5)  ∥H_{k}(x, .)∥_{L^{q}(Ω_{k})} ≤ Cd(x, ∂Ω_{k})^{4−n+ \frac{n}{q}} .

The constant C can be chosen uniformly for the family (Ω_{k})_{k∈ℕ}.

Proof. We proceed with the help of a duality argument. Let ψ ∈ C_{c}^{∞}(Ω_{k}) and let φ ∈ C^{4,α}(\overline{Ω_{k}}) be a solution of

\{ \begin{matrix}Δ^{2}φ = ψ & in Ω_{k}, \\ φ = ∂_{ν}φ = 0 & on ∂Ω_{k}. \end{matrix}

Let q ∈ (\frac{n}{n−3}, \frac{n}{n−4}) and denote q′ = \frac{q}{q−1} the dual exponent, so that in particular \frac{n}{4} < q′ < \frac{n}{3}. It follows from elliptic estimates [1, Thm. 15.2] that there exists C_{3} > 0 independent of φ, ψ and k such that

∥φ∥_{W^{4,q′} (Ω_{k})} ≤ C_{3}∥ψ∥_{L^{q′} (Ω_{k})}.

The embedding W^{4,q′}(Ω_{k}) ↪ C^{0,β}(\overline{Ω_{k}}) with β = 4 − \frac{n}{q′} = 4 − n + \frac{n}{q} being continuous uniformly in k shows that there exists C_{4} > 0 independent of φ and k such that ∥φ∥_{C^{0,β}(\overline{Ω_{k}})} ≤ C_{4}∥φ∥_{W^{4,q′} (Ω_{k})}. Let x ∈ Ω_{k} and x′ ∈ ∂Ω_{k}. We then get that

|φ(x)| = |φ(x) − φ(x′)| ≤ ∥φ∥_{C^{0,β}(\overline{Ω_{k}})}|x − x′|^{β} ≤ C_{3}C_{4}∥ψ∥_{L^{q′} (Ω_{k})}|x − x′|^{β}.

Moreover, it follows from Green’s representation formula that φ(x) = ∫_{Ω_{k}} H_{k}(x, y)ψ(y) dy for all x ∈ Ω_{k}. Therefore, taking the infimum with respect to x′ ∈ ∂Ω_{k}, we have that

|∫_{Ω_{k}} H_{k}(x, y)ψ(y) dy| ≤ C_{3}C_{4}∥ψ∥_{L^{q′} (Ω_{k})}d(x, ∂Ω_{k})^{β}

for all ψ ∈ C_{c}^{∞}(Ω_{k}). Inequality (4.5) then follows. □

Lemma 4.2. Assuming n > 4 and (4.3), one has that lim_{k→+∞} |x_{k} − y_{k}| = 0.

Proof. Assume by contradiction that |x_{k} − y_{k}| does not converge to 0. After extracting a subsequence we may then assume that there exists δ > 0 such that all x_{k} ∈ B_{δ}(x_{∞}) and all y_{k} ∈ Ω_{k} \ \overline{B_{3δ}(x_{∞})}. We consider q as in Lemma 4.1. In particular we know that ∥H_{k}(x, .)∥_{L^{q}(Ω_{k})} ≤ C uniformly in k. By applying local elliptic estimates (cf. [1, Theorem 15.3]) combined with Sobolev embeddings in Ω_{k} \ \overline{B_{2δ}(x_{∞})} we find that

∥H_{k}(x_{k}, .)∥_{L^{∞}(Ω_{k}\\overline{B_{3δ}(x_{∞})})} ≤ C(q, δ)

uniformly in k. In particular, we would have

|H_{k}(x_{k}, y_{k})| ≤ C(q, δ) and |x_{k} − y_{k}|^{n−4}|H_{k}(x_{k}, y_{k})| ≤ C(q, δ)

independent of k. This contradicts our hypothesis (4.3). □

Concluding the proof of Theorem 4, case n > 4. In what follows we may work in one fixed coordinate domain U_{i}; for this reason we drop the index i. Let Φ_{k} : U → ℝ^{n} be coordinate charts of Ω_{k} at x_{∞} as in Definition 1. We recall that

Φ_{k}(U ∩ {x_{1} < 0}) = Φ_{k}(U) ∩ Ω_{k} and Φ_{k}(U ∩ {x_{1} = 0}) = Φ_{k}(U) ∩ ∂Ω_{k}.

Without loss of generality we may assume that Φ_{k}(0) = x_{∞} and B_{δ}(x_{∞}) ⊂ Φ_{k}(U).

We let x_{k} = Φ_{k}(x′_{k}) and y_{k} = Φ_{k}(y′_{k}). Therefore, (4.4) rewrites as

(4.6)  lim_{k→+∞} x′_{k} = 0 and lim_{k→+∞} \frac{x′_{k,1}}{|x′_{k} − y′_{k}|} = 0.

We define for R large enough

H̃_{k}(z) = |x′_{k} − y′_{k}|^{n−4}H_{k}(Φ_{k}(x′_{k}), Φ_{k}(x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})))

in B_{R}(0) ∩ \{x_{1} < 0\}, where ρ_{k} := \frac{x′_{k,1}}{|x′_{k}−y′_{k}|}. The biharmonic equation Δ^{2}H_{k}(x, .) = 0 complemented with Dirichlet boundary conditions is rewritten as

Δ_{g_{k}}^{2} H̃_{k} = 0 in (B_{R}(0) ∩ {z_{1} < 0}) \ {ρ_{k}e⃗_{1}},  H̃_{k} = ∂_{1}H̃_{k} = 0 on {z_{1} = 0}.

Here, g_{k}(z) = Φ_{k}^{∗}(E)(x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})), E = (δ_{ij}) the Euclidean metric, and Δ_{g_{k}} denotes the Laplace-Beltrami operator with respect to this rescaled and translated pull back of the Euclidean metric under Φ_{k}. Then, for τ > 0 being chosen suitably small, it follows from elliptic estimates (see [1, Theorem 15.3]) and Sobolev embeddings that there exists C(R, q, τ) > 0 such that

(4.7)  |H̃_{k}(z)| ≤ C(R, q, τ)∥H̃_{k}∥_{L^{q}(B_{R}(0)\B_{τ} (0))}

for all z ∈ B_{R/2}(0) \ B_{2τ} (0), z_{1} ≤ 0. In order to estimate the L^{q}-norm on the right-hand side we use (4.5) and obtain that

∫_{B_{R}(0)∩{ζ_{1}<0}} |H̃_{k}(ζ)|^{q} dζ ≤ C|x′_{k} − y′_{k}|^{q(n−4)−n} ∫_{Ω_{k}} |H_{k}(x_{k}, y)|^{q} dy
≤ C|x′_{k} − y′_{k}|^{q(n−4)−n}d(x_{k}, ∂Ω_{k})^{(4−n)q+n}
≤ C (\frac{d(x_{k}, ∂Ω_{k})}{|x′_{k} − y′_{k}|})^{n−q(n−4)} .

Therefore, with (4.4), we get that lim_{k→+∞} ∥H̃_{k}∥_{L^{q}(B_{R}(0)\B_{τ} (0))} = 0, and (4.7) yields

lim_{k→+∞} H̃_{k} = 0 in C^{0}((B_{R/2}(0) \ B_{2τ} (0)) ∩ {z_{1} ≤ 0}).

In particular, since lim_{k→+∞} ρ_{k} = 0, we have that

lim_{k→+∞} H̃_{k} (\frac{y′_{k} − x′_{k}}{|y′_{k} − x′_{k}|} + ρ_{k}e⃗_{1}) = 0.

This limit rewrites as

lim_{k→+∞} |x_{k} − y_{k}|^{n−4}|H_{k}(x_{k}, y_{k})| = 0,

contradicting (4.3). The proof of Theorem 4, n > 4, is complete. □

Proof of Theorem 4, case n = 4. Here it is enough to prove (4.2) for ∇_{y}, exploiting the symmetry of the Green’s function. We argue by contradiction and as in the proof for n > 4, we may assume that there exist two sequences (x_{k})_{k∈ℕ}, (y_{k})_{k∈ℕ} with x_{k}, y_{k} ∈ Ω_{k} such that x_{k} ≠ y_{k} and

(4.8)  lim_{k→+∞} |x_{k} − y_{k}| |∇_{y}H_{k}(x_{k}, y_{k})| = +∞.

After possibly passing to a subsequence, it follows from (3.3) that there exists x_{∞} ∈ ∂Ω such that

(4.9)  lim_{k→+∞} x_{k} = x_{∞} and lim_{k→+∞} \frac{d(x_{k}, ∂Ω_{k})}{|x_{k} − y_{k}|} = 0.

Lemma 4.1 may be applied with some q > 4. The analogue of Lemma 4.2 is proved in exactly the same way as above. Like above we now put for R large enough

H̃_{k}(z) = H_{k}(Φ_{k}(x′_{k}), Φ_{k}(x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})))

in B_{R}(0) ∩ \{z_{1} < 0\}, where x_{k} = Φ_{k}(x′_{k}), y_{k} = Φ_{k}(y′_{k}), ρ_{k} := \frac{x′_{k,1}}{|x′_{k}−y′_{k}|}. As above we find for τ > 0 small enough that there exists C(R, τ, q) > 0 such that

|∇H̃_{k}(z)| ≤ C(R, q, τ)∥H̃_{k}∥_{L^{q}(B_{R}(0)\B_{τ} (0))}

for all z ∈ B_{R/2}(0) \ B_{2τ} (0), z_{1} ≤ 0. Using (4.5) we obtain that

∫_{B_{R}(0)∩{ζ_{1}<0}} |H̃_{k}(ζ)|^{q} dζ ≤ C|x′_{k} − y′_{k}|^{−4} ∫_{Ω_{k}} |H_{k}(x_{k}, y)|^{q} dy

≤ C (\frac{d(x_{k}, ∂Ω_{k})}{|x′_{k} − y′_{k}|})^{4} .

In the same way as in the generic case n > 4, this yields first that

lim_{k→+∞} ∇H̃_{k} = 0 in C^{0}((B_{R/2}(0) \ B_{2τ} (0)) ∩ {z_{1} ≤ 0})

and back in the original coordinates

lim_{k→∞} |x_{k} − y_{k}| |∇_{y}H_{k}(x_{k}, y_{k})| = 0.

So, we achieve a contradiction also if n = 4. This proves (4.2). Integrating (4.2), we get (4.1). The proof of Theorem 4 is complete. □

5. Convergence of the Green’s functions

As before, we consider a family of bounded regular domains (Ω_{k}) being a C^{4,α}-smooth perturbation of a fixed bounded C^{4,α}-smooth domain Ω according to Definition 1. We consider the operators Δ^{2} + a_{k} in Ω_{k} and assume that

∃U_{0} ⊃ \overline{Ω_{k}} : a_{k} ∈ C^{0,α}(U_{0});

∃a_{∞} ∈ C^{0,α}(U_{0}) : lim_{k→∞} a_{k} = a_{∞} in C_{loc}^{0,α}(U_{0}).

As before, we denote by G_{k} the Green’s functions corresponding to Δ^{2}+a_{k} in Ω_{k} and by G the Green’s functions corresponding to Δ^{2} + a_{∞} in Ω and show the following convergence result. As for the diffeomorphisms Φ_{k,i}, Φ_{i} we refer to Definition 1.

Proposition 5.1. Let x_{k} ∈ Ω_{k} and assume that lim_{k→∞} x_{k} = x_{∞} ∈ Ω. Then, we have:

G_{k}(x_{k}, .) → G(x_{∞}, .) in C_{loc}^{4}(Ω \ {x_{∞}}),
G_{k}(x_{k}, .) → G(x_{∞}, .) in L^{1}(ℝ^{n}),
G_{k}(x_{k}, .) ∘ Φ_{k,i} → G(x_{∞}, .) ∘ Φ_{i} in C_{loc}^{4}(U_{i} ∩ {z_{1} ≤ 0} \ {Φ_{i}^{−1}(x_{∞})}).

If n = 3 we have in addition that

G_{k}( . , .) → G( . , .) in C_{loc}^{0}(Ω × Ω).

Proof. According to Theorem 4 and Proposition 3.2 we know that (5.1)

|H_{k}(x, y)| ≤ C · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4, \\ 1, & if n = 3; \end{matrix} |G_{k}(x, y)| ≤ C · \{ \begin{matrix}|x − y|^{4−n}, & if n > 4, \\ (1 + |log |x − y||) , & if n = 4, \\ 1, & if n = 3; \end{matrix}

uniformly in k. This shows that in particular

(5.2)  ∥G_{k}(x, .)∥_{L^{1}(Ω_{k})} ≤ C uniformly in k.

Moreover, since x_{k} → x_{∞}, we may assume that all x_{k} are in a small neighbourhood around x_{∞}. Refering to the construction in the proof of Proposition 3.1 we see that the u_{k,x_{k}} arising there are uniformly bounded in C^{4,α} (\overline{Ω_{k}}). After selecting a suitable subsequence we see that for each Ω_{0} ⊂⊂ Ω one has G_{k}(x_{k}, .) → φ in C_{loc}^{4} (\overline{Ω_{0}} \ \{x_{∞}\}) and G_{k}(x_{k}, .) ∘ Φ_{k,i} → φ ∘ Φ_{i} in C_{loc}^{4}(U_{i} ∩ \{z_{1} ≤ 0\} \ \{Φ_{i}^{−1}(x_{∞})\}) with a suitable φ ∈ C^{4,α}(\overline{Ω}\\{x_{∞}\}). Thanks to this compactness and the fact that in any case the limit is the uniquely determined Green’s function, we have convergence on the whole sequence towards G(x_{∞}, .).

Finally, since we have pointwise convergence, (5.1) allows for applying Vitali’s convergence theorem to show that

G_{k}(x_{k}, .) → G(x_{∞}, .) in L^{1}(ℝ^{n}).

The statement concerning C_{loc}^{0}(Ω × Ω)-convergence in n = 3 follows from |∇G_{k}( . , .)| ≤ C, cf. (3.3). □

In order to prove Lemma 6.4 below we also need a convergence result simultaneous in both variables.

Proposition 5.2. We have that

G_{k}( . , .) ∘ (Φ_{k,i} × Φ_{k,j}) → G( . , .) ∘ (Φ_{i} × Φ_{j}) in C_{loc}^{4} ((U_{i} ∩ {x_{1} ≤ 0}) × (U_{j} ∩ {x_{1} ≤ 0}) \ D_{ij}) ,

where

D_{ij} = {(x, y) ∈ U_{i} × U_{j} : Φ_{i}(x) = Φ_{j}(y)}.

Proof. We combine the ideas of the proofs of Propositions 5.1 and 3.3. One should observe that Theorem 4 and Proposition 3.2 guarantee uniform L^{1}-bounds for H_{k} and G_{k} as in the proof of Proposition 5.1. □

6. The limit of the zeros of the Green’s functions

We keep the notations of the previous sections. In order to prove Theorem 3, we assume that for every k, there exist

(6.1)  x_{k}, y_{k} ∈ Ω_{k}, x_{k} ≠ y_{k} :  G_{k}(x_{k}, y_{k}) = 0.

After passing to subsequences there exist x_{∞} = lim_{k→∞} x_{k}, y_{∞} = lim_{k→∞} y_{k}. Using Definition 1, one sees that x_{∞}, y_{∞} ∈ \overline{Ω}.

As for the location of these limit points, we distinguish several cases.

6.1. Both points in the interior. Here, we consider the case that x_{∞}, y_{∞} ∈ Ω. Once it is shown that x_{∞} ≠ y_{∞} we conclude directly from Proposition 5.1 that

(6.2)  G(x_{∞}, y_{∞}) = 0.

So, we are left with proving:

Lemma 6.1. x_{∞} ≠ y_{∞}.

Proof. Assume by contradiction that x_{∞} = y_{∞}. We consider first the case n > 4 and here, the rescaled Green’s function:

(6.3)  G̃_{k}(z) := |x_{k} − y_{k}|^{n−4}G_{k}(x_{k}, x_{k} + |x_{k} − y_{k}|z).

Let ε > 0 be such that \overline{B_{2ε}(x_{∞})} ⊂ Ω ∩ Ω_{k} for all k. Then, for k large enough, |x_{k} − x_{∞}| < ε and G̃_{k}(z) is certainly defined for |z| < \frac{ε}{|x_{k}−y_{k}|}, where one has by Theorem 4 and Proposition 3.2 that

(6.4)  |G̃_{k}(z)| ≤ C|z|^{4−n}

uniformly in k. Because the G̃_{k} are defined on a sequence of sets which exhausts the whole ℝ^{n} we may discuss how to pass to a limit locally in ℝ^{n}. Since

Δ^{2}G̃_{k} + |x_{k} − y_{k}|^{4}a_{k}(x_{k} + |x_{k} − y_{k}|z)G̃_{k} = 0 on \overline{B_{ε/|x_{k}−y_{k}|}(0)} \ \{0\},

by elliptic Schauder theory we may assume that after possibly passing to a subsequence that

(6.5)  G̃_{k} → G̃ in C_{loc}^{4}(ℝ^{n} \ {0}), where |G̃(z)| ≤ C|z|^{4−n}.

Moreover,

Δ^{2}G̃ = 0 in ℝ^{n} \ {0}.

In order to compute the differential equation satisfied by G̃ near z = 0, let φ ∈ C_{c}^{∞}(ℝ^{n}) with supp φ ⊂ B_{R}(0) and define for k large enough

Ω_{k} ∋ x ↦ φ_{k}(x) := φ (\frac{x − x_{k}}{|x_{k} − y_{k}|}) , φ_{k} ∈ C_{c}^{∞}(Ω_{k}).

φ(0) = φ_{k}(x_{k}) = ∫_{Ω_{k}} G_{k}(x_{k}, y)(Δ^{2}φ_{k} + a_{k}φ_{k}) dy
= ∫_{B_{|x_{k}−y_{k}|R}(x_{k})} G_{k}(x_{k}, y)|x_{k} − y_{k}|^{−4} ((Δ^{2}φ) (\frac{y − x_{k}}{|x_{k} − y_{k}|}) + |x_{k} − y_{k}|^{4}a_{k}(y)φ (\frac{y − x_{k}}{|x_{k} − y_{k}|})) dy
= ∫_{B_{R}(0)} |x_{k} − y_{k}|^{n−4}G_{k}(x_{k}, x_{k} + |x_{k} − y_{k}|z) (Δ^{2}φ(z) + |x_{k} − y_{k}|^{4}a_{k}(x_{k} + |x_{k} − y_{k}|z)φ(z)) dz
= ∫_{ℝ^{n}} G̃_{k}(z) (Δ^{2}φ(z) + |x_{k} − y_{k}|^{4}a_{k}(x_{k} + |x_{k} − y_{k}|z)φ(z)) dz.

We put γ_{n} = \frac{1}{2(n−4)(n−2)ne_{n}} and obtain, letting k → ∞:

∫_{ℝ^{n}} G̃(z)Δ^{2}φ(z) dz = φ(0) = ∫_{ℝ^{n}} γ_{n}|z|^{4−n}Δ^{2}φ(z) dz.

This shows that we have in the sense of distributions that

Δ^{2} (G̃(z) − γ_{n}|z|^{4−n}) = 0 in ℝ^{n}.

Hence,

G̃(z) = γ_{n}|z|^{4−n} + ψ(z), ψ ∈ C^{∞}(ℝ^{n}), Δ^{2}ψ = 0.

Thanks to (6.4) we know further that

|ψ(z)| ≤ C(1 + |z|)^{4−n}.

Also for entire bounded biharmonic (even more generally polyharmonic) functions, Liouville’s theorem holds true, i.e. these are constant, see [23, p. 19]. Hence ψ(z) ≡ 0 showing that

G̃(z) = γ_{n}|z|^{4−n}, z ∈ ℝ^{n} \ {0}.

On the other hand we have according to the choice (6.1) of x_{k}, y_{k} and the definition (6.3) of G̃_{k} that

G̃_{k} (\frac{y_{k} − x_{k}}{|x_{k} − y_{k}|}) = |x_{k} − y_{k}|^{n−4}G_{k} (x_{k}, y_{k}) = 0.

Hence there exists at least one point ζ ∈ ℝ^{n} with

|ζ| = 1 and 0 = G̃(ζ) = γ_{n}|ζ|^{4−n},

which is false. This proves the statement for the case n > 4. One should observe that when looking just at the biharmonic operator, a proof for the previous lemma would directly follow from the local positivity results in general domains, which are proved in [15]. This observation will be useful in what follows.

Let us now consider the case n = 4. Since x_{∞} ∈ Ω, according to [15], there exists (a small) δ_{1} > 0 such that for all k and all x, y ∈ Ω_{k} we have that

(6.6)  x, y ∈ B_{δ_{1}}(x_{∞}) ⇒ H_{k}(x, y) ≥ −\frac{1}{c_{3}} log |x − y|.

We estimate the difference between G_{k} and H_{k}. For arbitrary but fixed x ∈ Ω, we have that with respect to the y-variable, (H_{k} − G_{k}) (x, .) solves the following Dirichlet problem:

\{ \begin{matrix}Δ_{y}^{2} (H_{k} − G_{k}) (x, y) + a_{k}(y) (H_{k} − G_{k}) (x, y) = a_{k}(y)H_{k}(x, y), & y ∈ Ω_{k} \\ (H_{k} − G_{k}) (x, y) = \frac{∂}{∂ν_{y}} (H_{k} − G_{k}) (x, y) = 0, & y ∈ ∂Ω_{k}. \end{matrix}

According to Theorem 4, we have that ∥a_{k}( .)H_{k}(x, .)∥_{L^{2}(Ω_{k})} ≤ c_{4} uniformly in k and x. Since Δ^{2} + a_{k} is assumed to be uniformly coercive, elliptic estimates from [1] show that

∥ (H_{k} − G_{k}) (x, .)∥_{L^{∞}(Ω_{k})} ≤ C∥ (H_{k} − G_{k}) (x, .)∥_{W^{4,2}(Ω_{k})} ≤ c_{5},

uniformly in x and k. Together with (6.6), this gives that there exist a δ_{2} > 0 and a constant c_{6} > 0 such that

(6.7)  x, y ∈ B_{δ_{2}}(x_{∞}) ⇒ G_{k}(x, y) ≥ −\frac{1}{c_{6}} log |x − y|.

This proves the claim also for n = 4, since by (6.7), it is impossible that G_{k}(x_{k}, y_{k}) = 0, where x_{k}, y_{k} → x_{∞} ∈ Ω.

Finally, we consider n = 3. Since here, according to Proposition 5.1, also G_{k} → G in C_{loc}^{0}(Ω × Ω) we have by assumption that G(x_{∞}, x_{∞}) = 0. On the other hand, testing the boundary value problem for G(x_{∞}, .) with G(x_{∞}, .) itself yields by virtue of the uniform coercivity that

G(x_{∞}, x_{∞}) ≥ λ ∫_{Ω} G(x_{∞}, y)^{2} dy > 0.

We obtain a contradiction also in the case n = 3. So, the proof of Lemma 6.1 is complete. □

6.2. One point in the interior, one point on the boundary. After possibly interchanging x_{∞} and y_{∞} we may consider the case that x_{∞} ∈ Ω, y_{∞} ∈ ∂Ω.

Lemma 6.2.

Δ_{y}G(x_{∞}, y_{∞}) = 0.

Proof. We may fix a neighbourhood B_{δ}(p_{i}) such that y_{∞} ∈ ∂Ω ∩ B_{δ}(p_{i}) so that for k large enough y_{k} ∈ Ω_{k} ∩ B_{δ}(p_{i}). We denote y′_{k} := Φ_{k,i}^{−1}(y_{k}), y′_{∞} := Φ_{i}^{−1}(y_{∞}) and observe that (y′_{k})_{1} < 0, (y′_{∞})_{1} = 0, y′_{k} → y′_{∞} in U_{i}. we recall the notation y′_{k} = (y′_{k,1}, ȳ′_{k}). Writing

G̃_{k,i} := G_{k}(x_{k}, .) ∘ Φ_{k,i}, G̃_{i} := G_{k}(x_{∞}, .) ∘ Φ_{i}

we see by means of Taylor’s expansion that with suitable θ_{k} ∈ (0, 1):

0 = G_{k}(x_{k}, y_{k}) = G̃_{k,i}(y′_{k})
= G̃_{k,i}(0, ȳ′_{k}) + ∂_{1}G̃_{k,i}(0, ȳ′_{k})y′_{k,1} + \frac{1}{2}∂_{11}G̃_{k,i}(θ_{k}y′_{k,1}, ȳ′_{k})(y′_{k,1})^{2}
= \frac{1}{2}∂_{11}G̃_{k,i}(θ_{k}y′_{k,1}, ȳ′_{k})(y′_{k,1})^{2}

due to the boundary conditions on G_{k}. According to Proposition 5.1 this yields ∂_{11}G̃_{i}(y′_{∞}) = 0. Since G_{k}(x_{k}, .)|_{∂Ω} = \frac{∂}{∂ν} G_{k}(x_{k}, .)|_{∂Ω} = 0, we obtain back in the original coordinates that Δ_{y}G(x_{∞}, y_{∞}) = 0 as stated. □

6.3. Both points on the boundary. So, here we have to consider the case that both x_{∞} ∈ ∂Ω and y_{∞} ∈ ∂Ω. The most delicate part will be to prove that both points have to be distinct:

Lemma 6.3. x_{∞} ≠ y_{∞}.

This statement is the core of the proof of Theorem 3; its proof is rather involved and will be postponed to Subsection 6.4. Assuming now Lemma 6.3 being proved it is not too difficult that in this case an additional zero of the Green’s function can be observed on the boundary:

Lemma 6.4. Δ_{x}Δ_{y}G(x_{∞}, y_{∞}) = 0.

Proof. According to Proposition 3.3 we have that G ∈ C^{4,α} in a neighbourhood of (x_{∞}, y_{∞}). This proof is similar to that of Lemma 6.2. We fix neighbourhoods such that x_{∞} ∈ B_{δ}(p_{i}), y_{∞} ∈ B_{δ}(p_{j}); without loss of generality we may assume that B_{δ}(p_{i}) ∩ B_{δ}(p_{j}) = ∅. Moreover we may assume that ∀k : x_{k} ∈ B_{δ}(p_{i}), y_{k} ∈ B_{δ}(p_{j}). To work in local charts we define

x′_{k} := Φ_{k,i}^{−1}(x_{k}), x′_{∞} := Φ_{i}^{−1}(x_{∞}), y′_{k} := Φ_{k,j}^{−1}(y_{k}), y′_{∞} := Φ_{j}^{−1}(y_{∞}).

Hence we have

x′_{k} ∈ U_{i} ∩ {x_{1} < 0}, x′_{k} → x′_{∞} ∈ U_{i} ∩ {x_{1} = 0}, y′_{k} ∈ U_{j} ∩ {y_{1} < 0}, y′_{k} → y′_{∞} ∈ U_{j} ∩ {y_{1} = 0}.

Defining

G̃_{k} : U_{i} ∩ {x_{1} ≤ 0} × U_{j} ∩ {y_{1} ≤ 0} → ℝ,  G̃_{k}(x′, y′) := G_{k} (Φ_{k,i}(x′), Φ_{k,j}(y′)) ;
G̃ : U_{i} ∩ {x_{1} ≤ 0} × U_{j} ∩ {y_{1} ≤ 0} → ℝ,  G̃(x′, y′) := G_{k} (Φ_{i}(x′), Φ_{j}(y′)) ;

we see that by assumption

0 = G_{k}(x_{k}, y_{k}) = G̃_{k}(x′_{k}, y′_{k}).

Taylor’s expansion with respect to y′ and exploiting the boundary conditions for G̃_{k} with respect to y′ shows that for each k there exists a suitable θ_{k} ∈ (0, 1) such that

∂_{y_{1}}^{2} G̃_{k}(x′_{k,1}, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k}) = 0.

Now, we use Taylor’s expansion for this expression with respect to x′ and obtain with suitable τ_{k} ∈ (0, 1):

0 = ∂_{y_{1}}^{2} G̃_{k}(x′_{k,1}, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k})
= ∂_{y_{1}}^{2} G̃_{k}(0, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k}) + ∂_{x_{1}}∂_{y_{1}}^{2} G̃_{k}(0, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k})x′_{k,1}

+\frac{1}{2}∂_{x_{1}}^{2}∂_{y_{1}}^{2} G̃_{k}(τ_{k}x′_{k,1}, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k})(x′_{k,1})^{2}

= \frac{1}{2}∂_{x_{1}}^{2}∂_{y_{1}}^{2} G̃_{k}(τ_{k}x′_{k,1}, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k})(x′_{k,1})^{2}

so that

∂_{x_{1}}^{2}∂_{y_{1}}^{2} G̃_{k}(τ_{k}x′_{k,1}, x̄′_{k}, θ_{k}y′_{k,1}, ȳ′_{k}) = 0.

Since by Proposition 5.2 we have C^{4} convergence of G̃_{k} to G̃ it follows that

∂_{x_{1}}^{2}∂_{y_{1}}^{2} G̃(x′_{∞}, y′_{∞}) = 0.

Taking into account the boundary conditions of G and of G̃, back in the original variables we see that

Δ_{x}Δ_{y}G(x_{∞}, y_{∞}) = 0

thereby proving the claim. □

6.4. Proof of Lemma 6.3. We assume by contradiction that lim_{k→∞} x_{k} = x_{∞} = y_{∞} = lim_{k→∞} y_{k}. We choose a neighbourhood B_{δ}(p_{i}) ∋ x_{∞} and may assume that ∀k : x_{k}, y_{k} ∈ B_{δ}(p_{i}) ∩ Ω_{k}. As before we introduce

x′_{k} := Φ_{k,i}^{−1}(x_{k}), y′_{k} := Φ_{k,i}^{−1}(y_{k}), x′_{∞} := Φ_{i}^{−1}(x_{∞})

so that we have

x′_{k}, y′_{k} ∈ U_{i} ∩ {x_{1} < 0}, x′_{k} → x′_{∞}, y′_{k} → x′_{∞} ∈ U_{i} ∩ {x_{1} = 0}.

We distinguish two further cases according to whether the distance between x_{k} and y_{k} converges faster to 0 than the distance of these points to the boundary or vice verca.

First case: |x_{k} − y_{k}| = o (max(d(x_{k}, ∂Ω_{k}), d(y_{k}, ∂Ω_{k}))). After possibly interchanging x_{k} and y_{k} and passing to a subsequence we may assume that

|x_{k} − y_{k}| = o(d(x_{k}, ∂Ω_{k})).

This case is much simpler than the second case below and quite similar to the case where both points converge in the interior treated in Subsection 6.1. Like there we treat the case n > 4 first. In this case, we consider the rescaled Green’s functions:

G̃_{k}(z) := |x_{k} − y_{k}|^{n−4}G_{k}(x_{k}, x_{k} + |x_{k} − y_{k}|z).

These are is certainly defined for |z| < \frac{d(x_{k},∂Ω_{k})}{|x_{k}−y_{k}|}, which converges to ∞ as k → ∞. For this reason, we may now directly copy the reasoning of Subsection 6.1 and obtain that

G̃_{k} → G̃ in C_{loc}^{4}(ℝ^{n} \ {0}) with G̃(z) = γ_{n}|z|^{4−n}.

One should observe that also here the property of the Green’s functions to be uniformly bounded by C|x−y|^{4−n} is used. According to the choice (6.1) of x_{k}, y_{k} and the definition of G̃_{k} we have that

G̃_{k} (\frac{y_{k} − x_{k}}{|x_{k} − y_{k}|}) = |x_{k} − y_{k}|^{n−4}G_{k} (x_{k}, y_{k}) = 0.

Hence there exists at least one point ζ ∈ ℝ^{n} with

|ζ| = 1 and 0 = G̃(ζ) = γ_{n}|ζ|^{4−n},

which is false.

We now treat the case n = 4 and proceed similarly as in the proof of Lemma 6.1. Rescaling the result of [15] shows the existence of δ > 0, c_{3} > 0 such that for x, y ∈ Ω_{k} with |x − y| ≤ δd(x, ∂Ω_{k}), one has (uniformly in k) that

(6.8)  H_{k}(x, y) ≥ −\frac{1}{c_{3}} log \frac{|x − y|}{d(x, ∂Ω_{k})}.

As it was shown in the proof of Lemma 6.1, G_{k} −H_{k} is bounded uniformly in k. Hence, there exists a constant c_{4} such that for x, y ∈ Ω_{k} we have

|x − y| ≤ δ d(x, ∂Ω_{k}) ⇒ G_{k}(x, y) ≥ −\frac{1}{c_{3}} log \frac{|x − y|}{d(x, ∂Ω_{k})} − c_{4}.

Since |x_{k} − y_{k}| = o(d(x_{k}, ∂Ω_{k})) we obtain

0 = G_{k}(x_{k}, y_{k}) → ∞ (k → ∞).

This is again false and proves the claim for n = 4.

Finally we discuss the case n = 3. Rescaling the result of Nehari [22] shows the existence of δ > 0, ε > 0 such that for x, y ∈ Ω_{k} with |x − y| ≤ δd(x, ∂Ω_{k}), one has (uniformly in k) that

(6.9)  H_{k}(x, y) ≥ εd(x, ∂Ω_{k}).

Making use of elliptic theory as in the proof of Lemma 6.1 and exploiting the fact that n = 3 yields that ∥ (G_{k}( . , y_{k}) − H_{k}( . , y_{k})) ∥_{C^{2}(\overline{Ω_{k}})} ≤ c_{5} uniformly in k. Since |x_{k} − y_{k}| ≤ δd(x_{k}, ∂Ω_{k}) for k large enough we conclude that

0 = G_{k}(x_{k}, y_{k}) ≥ εd(x_{k}, ∂Ω_{k}) − c_{6}d(x_{k}, ∂Ω_{k})^{2},

which becomes again false for k → ∞. (When discussing general polyharmonic operators (−Δ)^{m} the preceding argument would require the assumption n ≥ 2m − 1.)

Second case: |x_{k} − y_{k}| ≠ o(max(d(x_{k}, ∂Ω_{k}), d(y_{k}, ∂Ω_{k})). After selecting a subsequence we may assume that there is τ > 0 such that

|x_{k} − y_{k}| ≥ τd(x_{k}, ∂Ω_{k}) and |x_{k} − y_{k}| ≥ τd(y_{k}, ∂Ω_{k}).

We define

ρ_{k} := \frac{(x′_{k})_{1}}{|x′_{k} − y′_{k}|} < 0 and O(1)

and after selecting a further subsequence we may assume that

lim_{k→∞} ρ_{k} =: ρ ≤ 0.

Again, we will introduce a rescaled family of Green’s functions. For any R > 0 and z, ζ ∈ B_{R} ∩ ℝ_{−}^{n},

(6.10)  G̃_{k}(z, ζ) := |x′_{k} − y′_{k}|^{n−4}G_{k} (Φ_{k,i}(x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})), Φ_{k,i}(x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1}))) .

Moreover, G̃_{k}(z, ·) = ∂_{ζ_{1}} G̃_{k}(z, ·) = 0 on B_{R}(0) ∩ ∂ℝ_{−}^{n}. According to (4.1) and Proposition 3.2, we see that uniformly in k, z and ζ

(6.11)  |G̃_{k}(z, ζ)| ≤ C|z − ζ|^{4−n}, provided that n > 4.

If n = 3, 4 we conclude first that

|∇G̃_{k}(z, ζ)| ≤ C · \{ \begin{matrix}|z − ζ|^{−1}, & if n = 4, \\ 1, & if n = 3. \end{matrix}

Upon integration we obtain that

(6.12)  |G̃_{k}(z, ζ)| ≤ C · \{ \begin{matrix}(1 + | log |z − ζ|| + log(1 + |z|) + log(1 + |ζ|)) , & if n = 4, \\ (1 + |z| + |ζ|) , & if n = 3. \end{matrix}

The points x_{k} and y_{k} were chosen such that G_{k}(x_{k}, y_{k}) = 0, which reads in new coordinates

(6.13)  G̃_{k} (ρ_{k}e⃗_{1}, \frac{y′_{k} − x′_{k}}{|x′_{k} − y′_{k}|} + ρ_{k}e⃗_{1}) = 0.

In order to formulate the differential equation satisfied by G̃_{k}, we denote by E = (δ_{ij}) the Euclidean metric and

g_{k,i}(z) := Φ_{k,i}^{∗}(E)(x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1}))

its translated and rescaled pullback with respect to the coordinate charts Φ_{k,i}. Moreover, we introduce its limit, the constant metric

g_{∞,i} := Φ_{i}^{∗}(E)(x_{∞}).

First, we keep z ∈ ℝ_{−}^{n} fixed and consider G̃_{k}(z, .) =: G̃_{k,z}( .) as function in the second variable. For ζ ∈ B_{R}(0) ∩ ℝ_{−}^{n} \ {z} we have that for k large enough, the following boundary value problem is satisfied: (6.14)

\{ \begin{matrix}Δ_{g_{k,i},ζ}^{2} G̃_{k}(z, ζ) + |x′_{k} − y′_{k}|^{4}(a_{k} ∘ Φ_{k,i}) (x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1})) G̃_{k}(z, ζ) = 0 & for ζ_{1} < 0, ζ ≠ z, \\ G̃_{k}(z, ζ) = ∂_{ζ_{1}} G̃_{k}(z, ζ) = 0 & for ζ_{1} = 0. \end{matrix}

For k → ∞, using [1], we find G̃_{z} = G̃(z, .) ∈ C^{4} (\overline{ℝ_{−}^{n}} \ \{z\}) such that

(6.15)  G̃_{k}(z, .) → G̃_{z} in C_{loc}^{4} (\overline{ℝ_{−}^{n}} \ \{z\}) , Δ_{g_{∞},ζ}^{2} G̃(z, ζ) = 0 (z ≠ ζ);

(6.16)  |G̃(z, ζ)| ≤ C · \{ \begin{matrix}|z − ζ|^{4−n}, & if n > 4, \\ (1 + | log |z − ζ| | + log(1 + |z|) + log(1 + |ζ|) ) , & if n = 4, \\ (1 + |z| + |ζ|) , & if n = 3; \end{matrix}

(6.17)  |∇G̃(z, ζ)| ≤ C · \{ \begin{matrix}|z − ζ|^{−1}, & if n = 4, \\ 1, & if n = 3. \end{matrix}

In order to calculate the differential equation satisfied by G̃ near ζ = z, we introduce

φ ∈ C_{c}^{∞} (\overline{ℝ_{−}^{n}}) , φ = ∂_{1}φ = 0 on ∂ℝ_{−}^{n}

and let φ_{k} ∈ C^{4,α} (\overline{Ω_{k}}) such that

φ(z) = φ_{k} ∘ Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})) for z ∈ Ω̃_{k}
φ_{k} = ∂_{ν}φ_{k} = 0 on ∂Ω_{k};

where we denote

Ω̃_{k} := ρ_{k}e⃗_{1} − \frac{x′_{k}}{|x′_{k} − y′_{k}|} + \frac{1}{|x′_{k} − y′_{k}|} (U_{i} ∩ ℝ_{−}^{n}) .

By means of the representation formula and the corresponding Green’s function we see that for z ∈ ℝ_{−}^{n} and k large enough

φ(z) = φ_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})))
= ∫_{Ω_{k}} G_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})) , y) (Δ^{2}φ_{k} + a_{k}φ_{k}) dy
= ∫_{Φ_{k,i}(U_{i}∩{η_{1}<0})} G_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})) , y) (Δ^{2}φ_{k} + a_{k}φ_{k}) dy
= ∫_{U_{i}∩{η_{1}<0}} G_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(z − ρ_{k}e⃗_{1})) , Φ_{k,i}(η))
· (Δ_{Φ_{k,i}^{∗}(E)}^{2}(φ_{k} ∘ Φ_{k,i}) + (a_{k} ∘ Φ_{k,i})(φ_{k} ∘ Φ_{k,i})) (η) |Jac Φ_{k,i}(η)| dη
= ∫Ω̃_{k} |x′_{k} − y′_{k}|^{4−n}G̃_{k}(z, ζ)|x′_{k} − y′_{k}|^{−4} (Δ_{g_{k,i}}^{2} + |x′_{k} − y′_{k}|^{4}a_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1})))) φ(ζ)
·|x′_{k} − y′_{k}|^{n} |Jac Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1}))| dζ
= ∫Ω̃_{k} G̃_{k}(z, ζ) (Δ_{g_{k,i}}^{2} + |x′_{k} − y′_{k}|^{4}a_{k} (Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1})))) φ(ζ)
· |Jac Φ_{k,i} (x′_{k} + |x′_{k} − y′_{k}|(ζ − ρ_{k}e⃗_{1}))| dζ.

Observing (6.11), (6.12) and passing to the limit we obtain for z ∈ ℝ_{−}^{n}:

φ(z) = ∫_{ℝ_{−}^{n}} G̃(z, ζ)Δ_{g_{∞,i}}^{2}φ(ζ) |Jac Φ_{i}(x′_{∞})| dζ.

We introduce the linear bijection L = dΦ_{i}(x′_{∞}), the half space P := L (ℝ_{−}^{n}) and obtain for z ∈ ℝ_{−}^{n}:

(6.18)  φ(z) = ∫_{ℝ_{−}^{n}} G̃(z, ζ)Δ_{L^{∗}E}^{2}φ(ζ) |det(L)| dζ = ∫_{P} G̃(z, L^{−1}(η))Δ^{2} (φ ∘ L^{−1}) dη.

Finally we consider a rotation σ ∈ O(n) such that σ(P) = ℝ_{−}^{n} so that σ ∘ L(ℝ_{−}^{n}) = ℝ_{−}^{n}. For arbitrary

ψ ∈ C_{c}^{∞} (\overline{ℝ_{−}^{n}}) , with ψ = ∂_{1}ψ = 0 on ∂ℝ_{−}^{n}

and x̃ ∈ ℝ_{−}^{n} we may take φ = ψ ∘ σ ∘ L and z = (σ ∘ L)^{−1}(x̃). We obtain from (6.18) since the Laplacian is invariant under orthogonal transformations that for x̃ ∈ ℝ_{−}^{n}:

ψ(x̃) = (ψ ∘ σ ∘ L) ((σ ∘ L)^{−1}(x̃)) = ∫_{P =σ^{−1}(ℝ_{−}^{n})} G̃ ((σ ∘ L)^{−1}(x̃), L^{−1}(η)) Δ^{2} (ψ ∘ σ) (η) dη
= ∫_{ℝ_{−}^{n}} G̃ ((σ ∘ L)^{−1}(x̃), (σ ∘ L)^{−1}(η)) Δ^{2}ψ(η) dη.

This shows that in the sense of distributions

(6.19)  Δ2 ; ỹḠ(x̃, .) = δx̃,

where we have defined

(6.20)  Ḡ(x̃, ỹ) := G̃ ((σ ∘ L)^{−1}(x̃), (σ ∘ L)^{−1}(ỹ)) .

Moreover, for fixed x̃ ∈ ℝ_{−}^{n} one concludes with the help of (6.16) and (6.17) that

(6.21)  |Ḡ(x̃, ỹ)| ≤ C · \{ \begin{matrix}|x̃ − ỹ|^{4−n}, & if n > 4, \\ (1 + | log |x̃ − ỹ| | + log(1 + |x̃|) + log(1 + |ỹ|) ) , & if n = 4, \\ (1 + |x̃| + |ỹ|) , & if n = 3; \end{matrix}

(6.22)  |∇Ḡ(x̃, ỹ)| ≤ C · \{ \begin{matrix}|x̃ − ỹ|^{−1}, & if n = 4, \\ 1, & if n = 3. \end{matrix}

We denote by H the biharmonic Green’s function in ℝ_{−}^{n}, which thanks to Boggio [4] is known explicitly and known to be positive – see Lemma 6.6 below. We prove:

Lemma 6.5. ∀x, y ∈ ℝ_{−}^{n}, x ≠ y :  Ḡ(x, y) = H(x, y).

Proof. In what follows we keep x ∈ ℝ_{−}^{n} fixed. Both Ḡ(x, .) and H(x, .) satisfy the biharmonic equation with the δ-distribution δ_{x} as right hand side and zero Dirichlet boundary conditions on {y_{1} = 0}. We let ψ_{x} := Ḡ(x, .) − H(x, .). Hence,

ψ = ψ_{x} ∈ C^{∞} (\overline{ℝ_{−}^{n}})

solves

(6.23)  \{ \begin{matrix}Δ^{2}ψ = 0 & in ℝ_{−}^{n}, \\ ψ = ∂_{1}ψ = 0 & on \{y_{1} = 0\}. \end{matrix}

Moreover, according to (6.21-6.22) and (6.31) below we have that

(6.24)  ∀y ∈ ℝ_{−}^{n} :  |ψ(y)| ≤ C · \{ \begin{matrix}|y|^{4−n}, & if n > 4, \\ (1 + | log |y| |) , & if n = 4, \\ (1 + |y|) , & if n = 3; \end{matrix}

(6.25)  |∇ψ(y)| ≤ C · \{ \begin{matrix}|y|^{−1}, & if n = 4, \\ 1, & if n = 3; \end{matrix}

where C = C(x). According to [10, 18]

ψ^{∗}(y) := \{ \begin{matrix}ψ(y) & if y_{1} ≤ 0, \\ −ψ(−y_{1}, ȳ) − 2y_{1} \frac{∂}{∂y_{1}} ψ(−y_{1}, ȳ) − y_{1}^{2}Δψ(−y_{1}, ȳ), & if y_{1} > 0, \end{matrix}

ψ^{∗} ∈ C^{4} (ℝ^{n}) is an entire biharmonic function. We consider now first the case n > 4. Below we will prove that (6.23) and (6.24) imply that also

(6.26)  ∀j = 1, 2 : ∀y ∈ ℝ_{−}^{n} :  |∇^{j}ψ(y)| ≤ C|y|^{4−n−j}, where C = C(x).

This immediately gives that |ψ^{∗}(y)| ≤ C|y|^{4−n} and in particular that ψ^{∗} is a bounded entire biharmonic function. Again, Liouville’s theorem for biharmonic functions [23, p. 19] yields that ψ^{∗}(y) ≡ 0 so that the claim of the lemma follows, provided n > 4.

If n = 3, 4 we shall prove below that for j = 0, 1, 2

(6.27)  ∀y ∈ ℝ_{−}^{n} :  |D^{2+j}ψ(y)| ≤ C|y|^{2−n−j}, where C = C(x).

As above ψ^{∗} is an entire biharmonic function and so are Dψ^{∗} and D^{2}ψ^{∗}. Since |D^{2}ψ^{∗}(y)| ≤ C(1 + |y|)^{2−n}, it follows that D^{2}ψ^{∗}(x) ≡ 0. In view of the boundary conditions in (6.23) we come up with ψ^{∗}(y) ≡ 0 also in the case n = 3, 4.

It remains to prove (6.26) and (6.27). We consider first n > 4. Assume by contradiction that there exists a sequence (y_{ℓ}) ⊂ ℝ_{−}^{n} such that |∇^{j}ψ(y_{ℓ})| · |y_{ℓ}|^{n+j−4} → ∞ for ℓ → ∞. Then

ψ̃_{ℓ}(y) := |y_{ℓ}|^{n−4}ψ (y_{ℓ} − y_{ℓ,1}e⃗_{1} + |y_{ℓ}|y)

would solve

(6.28)  \{ \begin{matrix}Δ^{2}ψ̃_{ℓ} = 0 & in ℝ_{−}^{n}, \\ ψ̃_{ℓ} = ∂_{1}ψ̃_{ℓ} = 0 & on \{y_{1} = 0\}. \end{matrix}

From the assumption we conclude that

(6.29)  |∇^{j}ψ̃_{ℓ} (\frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1})| = |y_{ℓ}|^{n+j−4}|∇^{j}ψ(y_{ℓ})| → ∞.

On the other hand,

(6.30)  |ψ̃_{ℓ}(y)| ≤ C|y_{ℓ}|^{n−4} |y_{ℓ} − y_{ℓ,1}e⃗_{1} + |y_{ℓ}|y|^{4−n} ≤ C |\frac{y_{ℓ}}{|y_{ℓ}|} + y − \frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1}|^{4−n} ,

so that ψ̃_{ℓ} remains bounded in a neighbourhood of \frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1} in \overline{ℝ_{−}^{n}}. Local Schauder estimates [1, Theorem 7.3] yield

|∇^{j}ψ̃_{ℓ} (\frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1})| ≤ C,

thereby contradicting (6.29). This proves (6.26).

As for (6.27), i.e. in particular n = 3, 4, the proof is quite similar since we can already make use of the gradient estimates (6.25). Instead of (6.30) one has to make use of

|∇ψ̃_{ℓ}(y)| ≤ C|y_{ℓ}|^{n−3} |y_{ℓ} − y_{ℓ,1}e⃗_{1} + |y_{ℓ}|y|^{3−n} ≤ C |\frac{y_{ℓ}}{|y_{ℓ}|} + y − \frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1}|^{3−n} ,

so that ∇ψ̃_{ℓ} remains bounded uniformly outside \frac{y_{ℓ}}{|y_{ℓ}|} − \frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1}. Therefore, since ψ̃_{ℓ} vanishes on ∂ℝ_{−}^{n}, we get that ψ̃_{ℓ} is bounded in a neighbourhood of \frac{y_{ℓ,1}}{|y_{ℓ}|} e⃗_{1} in \overline{ℝ_{−}^{n}}. The proof of the present lemma is complete. □

In order to show that the present case x_{∞} = y_{∞} ∈ ∂Ω cannot occur we collect some basic facts on the biharmonic Green’s function in the half space; modulo a simple conformal transformation, cf. [4, p. 126]:

Lemma 6.6. The biharmonic Green’s function in ℝ_{−}^{n} is given by

(6.31)  ∀x, y ∈ ℝ_{−}^{n} : H(x, y) = \frac{1}{4ne_{n}} |x − y|^{4−n} ∫_{1}^{|x^{∗}−y|/|x−y|}(v^{2} − 1)v^{1−n} dv; x^{∗} = (−x_{1}, x̄).

From this it follows by direct calculation:

(6.32)  ∀x, y ∈ ℝ_{−}^{n}, x ≠ y :  H(x, y) > 0;
(6.33)  ∀x ∈ ℝ_{−}^{n}, y ∈ ∂ℝ_{−}^{n} :  Δ_{y}H(x, y) > 0;
(6.34)  ∀x, y ∈ ∂ℝ_{−}^{n}, x ≠ y :  Δ_{x}Δ_{y}H(x, y) > 0.

We proceed by showing that x_{∞} = y_{∞} ∈ ∂Ω is indeed impossible and recall that by assumption we chose x_{k}, y_{k} such that G_{k}(x_{k}, y_{k}) = 0. In terms of the transformed Green’s functions this reads

(6.35)  G̃_{k} (ρ_{k}e⃗_{1}, \frac{y′_{k} − x′_{k}}{|x′_{k} − y′_{k}|} + ρ_{k}e⃗_{1}) = 0,

cf. (6.13). After possibly extracting a further subsequence we find a point

θ = lim_{k→∞} \frac{y′_{k} − x′_{k}}{|x′_{k} − y′_{k}|}

and may conclude that

(6.36)  G̃ (ρe⃗_{1}, θ + ρe⃗_{1}) = 0.

According to the possible location of the limit points we have to distinguish four cases:

Case (a): ρ < 0 and (θ + ρe⃗_{1})_{1} < 0. We put x̃ = (σ ∘ L)(ρe⃗_{1}) ∈ ℝ_{−}^{n}, ỹ = (σ ∘ L)(θ + ρe⃗_{1}) ∈ ℝ_{−}^{n}. According to (6.20) and Lemma 6.5 we could conclude that

H(x̃, ỹ) = Ḡ(x̃, ỹ) = 0,

which is impossible in view of (6.32).

Case (b): ρ = 0 and (θ + ρe⃗_{1})_{1} < 0. As in the proof of Lemma 6.2 we conclude from (6.35) that ∂_{x_{1}}^{2} G̃(0, θ) = 0. Together with the Dirichlet boundary conditions satisfied by G̃ this yields G̃(0, θ) = 0, D_{x}G̃(0, θ) = 0, D_{x}^{2}G̃(0, θ) = 0. If we put ỹ = (σ ∘ L)(θ) ∈ ℝ_{−}^{n} this implies due to (6.20) that also D_{x}^{2}Ḡ(0, ỹ) = 0. In particular, we have that Δ_{x}H(0, ỹ) = Δ_{x}Ḡ(0, ỹ) = 0, which is impossible in view of (6.33).

Case (c): ρ < 0 and (θ + ρe⃗_{1})_{1} = 0. Due to symmetry of the Green’s function, this case is completely analogous to the previous one and hence impossible in view of (6.33).

Case (d): ρ = 0 and (θ + ρe⃗_{1})_{1} = 0. As in the proof of Lemma 6.4 we conclude from (6.35) that ∂_{x_{1}}^{2}∂_{y_{1}}^{2} G̃(0, θ) = 0. Here θ_{1} = 0, |θ| = 1. Thanks to the boundary conditions satisfied by G̃ this gives ∀|α| ≤ 2, |β| ≤ 2 : D_{x}^{α}D_{y}^{β}G̃(0, θ) = 0. Using again (6.20), we see that also ∀|α| ≤ 2, |β| ≤ 2 : D_{x}^{α}D_{y}^{β}Ḡ(0, ỹ) = 0, where ỹ = (σ ∘ L)(θ) ≠ 0. In particular, we come up with Δ_{x}Δ_{y}H(0, ỹ) = Δ_{x}Δ_{y}Ḡ(0, ỹ) = 0. This is impossible in view of (6.34).

Conclusion. In each case we finally deduced a contradiction so that x_{∞} = y_{∞} ∈ ∂Ω is indeed impossible. The proof of Lemma 6.3 is complete. □

6.5. Proof of Theorems 1, 2 and 3. .

Theorem 3 follows from the conclusions made in Subsections 6.1, 6.2 and 6.3.

In order to prove Theorem 1 we assume for contradiction that there exist a bounded C^{4,α}-smooth domain Ω ⊂ ℝ^{n} and sequences (x_{k}), (y_{k}) ⊂ Ω, x_{k} ≠ y_{k} with H_{Ω}(x_{k}, y_{k}) ≤ 0 and lim_{k→∞} |x_{k} − y_{k}| = 0. In view of the smoothness assumption, we see by working in local coordinate charts that after possibly passing to a subsequence and relabelling we find ỹ_{k} ∈ Ω, x_{k} ≠ ỹ_{k} with H_{Ω}(x_{k}, ỹ_{k}) = 0 and |x_{k} − ỹ_{k}| → 0 for k → ∞. Application of Theorem 3 in the special case Ω_{k} = Ω, a_{k} = 0 shows that this is impossible. This contradiction proves that there exists a δ = δ(Ω) > 0 such that x, y ∈ Ω, x ≠ y, H_{Ω}(x, y) ≤ 0 ⇒ |x − y| ≥ δ. Estimate (1.3) now follows directly from (4.1) while (1.4) is a consequence of (1.2), i.e. of DallAcqua and Sweers [8].

In order to prove Theorem 2, we assume that no such ε_{0} > 0 exists. In view of the remark after Theorem 2, we would have a neighbourhood U of \overline{B}, C^{4,α}-smooth diffeomorphisms ψ_{k} : \overline{U} → ψ_{k}(\overline{U}) and smooth domains Ω_{k} = ψ_{k}(B) with sign changing biharmonic Green’s functions H_{k}. Hence, one of the alternatives described in Theorem 3 would occur for the biharmonic Green’s function H in the ball B. Since H enjoys precisely the analogous properties of Lemma 6.6 (cf. [4, p. 126]), this is false; Theorem 2 follows. □

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[28] H.S. Shapiro and M. Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36, 99-101 (1994).

   E-mail address: hans-christoph.grunau@ovgu.de

Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany

   E-mail address: frobert@math.unice.fr

Université de Nice-Sophia Antipolis, Laboratoire J.A.Dieudonné, Parc Valrose, 06108 Nice Cedex 2, France