UNIFORM ESTIMATES FOR POLYHARMONIC GREEN FUNCTIONS IN DOMAINS WITH SMALL HOLES

HANS-CHRISTOPH GRUNAU AND FRÉDÉRIC ROBERT

Dedicated to Patrizia Pucci on the occasion of her 60th birthday.

Given an arbitrary C^{2k,θ}-smooth bounded domain Ω ⊂ ℝ^{n} with exterior unit normal ν, n > 2k ≥ 2 and θ ∈ (0, 1), we define G_{Ω} : \overline{Ω} × \overline{Ω} \ \{(x, x)/ x ∈ \overline{Ω}\} → ℝ as the Green function of (−Δ)^{k} in the domain Ω with Dirichlet boundary condition. This means that for f ∈ C^{0,θ}(\overline{Ω}) the unique solution u ∈ C^{2k,θ}(\overline{Ω}) of the polyharmonic Dirichlet problem

(1)  \{ \begin{matrix}(−Δ)^{k}u = f & in Ω, \\ u = ∂_{ν}u = … = ∂_{ν}^{(k−1)}u = 0 & on ∂Ω \end{matrix}

is given by

u(x) = ∫_{Ω} G_{Ω}(x, y)f(y) dy.

We are interested in pointwise estimates for G_{Ω}. In the special case k = 1, i.e. the case of the usual Laplacian, these can be deduced by using the maximum principle. This yields that G_{Ω} is positive und bounded from above by the fundamental solution, i.e. for n > 2 and any bounded smooth domain Ω ⊂ ℝ^{n} we have

(2)  ∀x, y ∈ Ω, x ≠ y,:  0 < G_{Ω}(x, y) < \frac{1}{(n − 2)ne_{n}} |x − y|^{2−n}.

Here, e_{n} denotes the measure of the n-dimensional unit ball. One should observe that the constant in the right inequality is independent of Ω, even with respect to singular perturbations.

When passing to biharmonic or –more general– polyharmonic equations, i.e. the cases k ≥ 2, the maximum principle is no longer available and positivity issues remain valid only in a very weak and modified sense. Mathematical contributions on this topic go back at least to Boggio and Hadamard [2,7]; these papers are also fundamental for subsequent works on estimating polyharmonic Green functions. For an extensive discussion of related and more recent contributions one may see the monograph Gazzola-Grunau-Sweers [3] and Grunau-Robert [5]. There is no obvious idea how to directly prove higher order analogues to estimate (2). However, basing on the general Schauder and L^{p}-theory developed by Agmon, Douglis, and Nirenberg [1], Krasovskiĭ [8,9] proved that for any given bounded sufficiently smooth domain Ω, there exists C_{Ω} > 0 such that

(3)  |G_{Ω}(x, y)| ≤ C_{Ω}|x − y|^{2k−n} for all x, y ∈ Ω, x ≠ y.

The constant C_{Ω} depends on C^{2k,θ}-properties of the boundary ∂Ω. In Krasovskiĭ’s works, very general operators and boundary conditions were discussed. Applying these general results to our special polyharmonic Dirichlet problems originally required a higher degree of smoothness. However, it turns out that for our purposes, C^{2k,θ}-smoothness of ∂Ω suffices. For more detailed information on this issue we refer to Theorem 2 in the appendix. Estimate (3) can also be extended to the derivatives of Green functions: For any 0 ≤ r ≤ 2k, there exists C_{Ω,r} such that

Date: September 28th 2012.

(4)  |∇_{y}^{r}G_{Ω}(x, y)| ≤ C_{Ω,r}|x − y|^{2k−n−r} for all x, y ∈ Ω, x ≠ y.

Here, ∇_{y}^{r} denotes any partial derivative with respect to y of order r.

The constant C_{Ω} in the Green function estimate (3) depends –as soon as k > 1– heavily on the smoothness properties of ∂Ω. As long as one considers families of domains with uniform smoothness properties one may choose the same constant. In the present article, we exhibit families of domains with unbounded curvature, namely fixed domains Ω where we punch out arbitrarily small holes. For uniform Green function estimates, (3) can no longer be used since the curvature blows-up, and so does the constant C(Ω). Nevertheless, we can prove the following uniform estimates.

Theorem 1. Let Ω be a C^{2k,θ}-smooth bounded domain of ℝ^{n} and let x_{0} ∈ Ω. Let ω be a C^{2k,θ}-smooth bounded domain of ℝ^{n} containing 0. We fix a number q ∈ (0, 1). Then there exists a constant C = C(Ω, ω, x_{0}, q) > 0 such that that for all ε ∈ (0, q \frac{d(x_{0},∂Ω)}{diam(ω)}), we have that

|G_{Ω_{ε}}(x, y)| ≤ C|x − y|^{2k−n} for all x, y ∈ Ω_{ε}, x ≠ y,

where Ω_{ε} := Ω \ \overline{\{x_{0} + εω\}}.

Remark 1. In small dimensions n ≤ 2k, a uniform estimate like (7) below is no longer available in the complements of arbitrarily small domains for i = 0. Nakai and Sario [11] discussed the biharmonic case k = 2 in dimension n = 2 with the help of energy estimates and their approach can probably be used for any k ≥ 2 and any dimension n < 2k. In this small dimensions case some (in general not all) of the Dirichlet boundary conditions remain in x_{0} even in the singular limit Ω_{0} = Ω \ {x_{0}}. This phenomenon cannot be expected in large dimensions n ≥ 2k.

It is then natural to ask whether in estimates like (4) we may also expect uniformity with respect to the family of domains (Ω_{ε})_{ε}. This, however, is not the case. More precisely, we have the following:

Proposition 1. Let Ω, q ∈ (0, 1), Ω_{ε}, ε > 0, be as in Theorem 1. Then for all 1 ≤ r ≤ 2k, we have that

sup_{ε∈(0,qd(x_{0},∂Ω)/ diam(ω))} sup_{x,y∈Ω_{ε}, x≠y} |x − y|^{n−2k+r}|∇_{y}^{r}G_{Ω_{ε}}(x, y)| = +∞.

As mentioned at the beginning, one has a comparison principle for (1) in general only in the second order case, i.e. if k = 1. In this case, G_{Ω} > 0 holds true for any Ω, while if k ≥ 2 one has positivity G_{Ω} > 0 only in very restricted classes of domains among which are balls (Boggio [2]) and small perturbations of balls (Grunau-Robert [5]). In general, however, one has sign change, i.e. G_{Ω} ≱ 0. Already Hadamard [7] observed that this will occur in the biharmonic case in twodimensional annuli with very small inner radii, see also Nakai-Sario [11]. On the other hand, for fixed domains, the negative part will be “relatively” small. For more detailed information on this issue one may see Grunau-Robert [5], Gazzola-Grunau-Robert [3] and Grunau-Robert-Sweers [6]. For instance, the authors proved in [5] that for any C^{4,θ}-smooth bounded domain Ω ⊂ ℝ^{n}, n > 4, there exists C(Ω) > 0 such that ∥(G_{Ω})_{−}∥_{L^{∞}(Ω)} ≤ C(Ω), where G_{Ω} is the Green function for (−Δ)^{2} with Dirichlet boundary condition. A natural question is to ask whether one may expect uniformity of this lower bound with respect to families of domains. As shown by the following proposition, the validity of this guess is equivalent to the nonnegativity of all Green functions:

Proposition 2. We assume that n > 2k. The two following assertions are equivalent:

(i) There exists C(k, n, θ) depending only on k, n, θ such that such that

∥(G_{Ω})_{−}∥_{L^{∞}(Ω)} ≤ C(k, n, θ)

for all C^{2k,θ}-smooth bounded domains Ω ⊂ ℝ^{n}. (ii) G_{Ω} ≥ 0 for all C^{2k,θ}-smooth bounded domains Ω ⊂ ℝ^{n}.

Since (ii) is false for the higher order case k ≥ 2 (see the discussion and references in the monograph Gazzola-Grunau-Sweers [3, pp. 62/63 and 69/70]) we conclude that there is no uniform bound for negative parts of biharmonic and polyharmonic Green functions. We emphasise that we only discuss Dirichlet boundary conditions and that positivity issues may be quite different for other boundary conditions.

Notation: In the sequel, C(a, b, …) denotes a constant depending on ω, Ω, a, b, . . .. The same notation can be used for two different constants from line to line, and even in the same line.

Proofs.

We start with proving Theorem 1 and proceed in several steps. In order to keep the exposition as simple as possible we shall prove the theorem for q = \frac{1}{42}. At the end of Step 3 we shall indicate how to modify the proof for larger q < 1. Without loss of generality, we assume that x_{0} = 0 so that Ω_{ε} := Ω \ εω.

Step 1. The Green function in the exterior domain ℝ^{n} \ ω.

Let ω be a C^{2k,θ} domain of ℝ^{n} such that 0 ∈ ω. We define

ω_{0} := inv(ℝ^{n} \ \overline{ω}) ∪ \{0\} , where inv : \{ \begin{matrix}ℝ^{n} \ \{0\} & → & ℝ^{n} \ \{0\}, \\ x & ↦ & \frac{x}{|x|^{2}} . \end{matrix}

We emphasise that inv is a special Möbius transform of ℝ^{n} and in particular conformal. The set ω_{0} is a C^{2k,θ}-smooth bounded domain of ℝ^{n} containing 0. We define

(5)  G_{(εω)^{c}}(x, y) := ε^{n−2k}|y|^{2k−n}|x|^{2k−n}G_{ω_{0}}(ε inv(x), ε inv(y))

for all x, y ∈ ℝ^{n} \ εω. The following proposition shows that this is indeed the polyharmonic Green function in (εω)^{c}:

Proposition 3. For any φ ∈ C_{c}^{2k}(ℝ^{n} \ εω) such that ∂_{ν}^{(i)}φ = 0 on ∂(εω) for i = 0, …, (k − 1), we have that

(6)  φ(x) = ∫_{ℝ^{n}\εω} G_{(εω)^{c}}(x, y)(−Δ)^{k}φ(y) dy

for all x ∈ ℝ^{n} \ εω. Moreover, for all 0 ≤ i ≤ 2k, the derivatives with respect to y satisfy the upper bound

(7)  |∇_{y}^{i}G_{(εω)^{c}}(x, y)| ≤ C|y|^{−i} ∑_{r≤i} |x|^{r}|x − y|^{2k−n−r}.

Proof. We prove the claim first for ε = 1. Let φ ∈ C_{c}^{2k}(ℝ^{n} \ ω) be such that ∂_{ν}^{(i)}φ = 0 on ∂ω for i = 0, …, (k − 1). We show that

(8)  φ(x) = ∫_{ℝ^{n}\ω} |y|^{2k−n}|x|^{2k−n}G_{ω_{0}}(inv(x), inv(y))(−Δ)^{k}φ(y) dy

for all x ∈ ℝ^{n} \ ω. Indeed, inv is the composition of two sterographic projections of opposite poles, and therefore, it is conformal and the pull-back of the Euclidean metric Eucl via inv is inv^{⋆} Eucl = | · |^{−4} Eucl = μ^{4/(n−2k)} Eucl where μ(x) := |x|^{2k−n} for all x ∈ ℝ^{n} \ {0}. As a consequence, considering (−Δ)^{k} as the conformal operator of Graham-Jenne-Mason-Sparling for the Euclidean space (see [4]), the conformal law of the GJMS operators yields

((−Δ)^{k}φ) ∘ inv = μ^{−(n+2k)/(n−2k)}(−Δ)^{k}(μ(φ ∘ inv)).

In addition, the Jacobian of inv and then the Riemannian element of volume of inv^{⋆} Eucl are

Jac(inv) = | · |^{−2n} and dv_{inv}^{⋆} _{Eucl} = | · |^{−2n} dx.

This transformation behaviour of polyharmonic operators with respect to Möbius transforms is classical, see e.g. Loewner [10] and references therein. A convenient and easily accessible reference is also Gazzola-Grunau-Sweers [3, Lemma 6.14] .

We fix x ∈ ℝ^{n}\\overline{ω} and we consider x′ := inv(x) ∈ ω_{0}\\{0\}. We define φ̃(y) := μ(y)φ∘ inv(y) = |y|^{2k−n}φ(y/|y|^{2}) for y ∈ \overline{ω}_{0}\\{0\}. We find that φ̃ is vanishing around 0 and therefore extends smoothly to \overline{ω}_{0}. It follows from Green’s representation formula that

φ̃(x′) = ∫_{ω_{0}} G_{ω_{0}}(x′, y)(−Δ)^{k}φ̃(y) dy.

Performing the change of variable y = inv(z) and using the above properties yields

φ̃(x′) = ∫_{ℝ^{n}\ω} |z|^{n+2k}G_{ω_{0}}(x′, inv(z))(−Δ)^{k}φ(z)|z|^{−2n} dz.

Going back to the expression of φ yields (8).

Given α a multi-index and j ∈ {1, …, n}, there exists an homogeneous polynomial P_{j}^{α} of degree |α| + 1 such that

∂^{α} inv(x)_{j} = \frac{P_{j}^{α}(x)}{|x|^{2(|α|+1)}}

for all x ∈ ℝ^{n} \ {0}, where |α| is the length of the index. We fix x, y ∈ ℝ^{n} \ ω such that x ≠ y. With help of the binomial formula, the derivative of order α with respect to y is such that

(9) |∂_{y}^{α}G_{ω^{c}}(x, y)| ≤ C|x|^{2k−n} ∑_{β≤α} |y|^{2k−n−|α|+|β|}|∂_{y}^{β}(G_{ω_{0}}(inv(x), inv(y)))|,

where we have adopted the standard order on multi-indices. For |β| ≥ 1, the chain rule yields

∂^{β}(f ∘ inv) = ∑_{1≤r≤|β|} ∑_{I_{1}+...+I_{r}=β} ∑_{j_{1},...,j_{r}} c_{j_{1},...,j_{r}}^{(I_{1},...,I_{r})}∂^{I_{1}} inv_{j_{1}} … ∂^{I_{r}} inv_{j_{r}}(∂^{j_{1}...j_{r}}f) ∘ inv

for any function f when the derivatives make sense. The second sum is taken over all decompositions of β as a sum of r multi-indices and the c_{j_{1},...,j_{r}}^{(I_{1},...,I_{r})} are combinatorial constants which can be calculated explicitly. When restricting to suitable decompositions of β these constants are equal to 1. This formula yields

(10)  |∂_{y}^{β}(G_{ω_{0}}(inv(x), inv(y)))| ≤ C ∑_{r≤|β|} |y|^{−|β|−r}|(∇^{r}G_{ω_{0}})(inv(x), inv(y))|

for all β ≤ α. Here, ∇^{r}f = (∂^{γ}f)_{|γ|=r} when this makes sense.

It follows from Krasovskiĭ [8, 9] that for any 0 ≤ r ≤ 2k, there exists C = C(ω_{0} = inv(\overline{ω}^{c}) ∪ \{0\}, r) > 0 such that

|∇^{r}G_{ω_{0}}(x, y)| ≤ C|x − y|^{2k−n−r}

for all x, y ∈ ω_{0}, x ≠ y. For the sake of completeness, we refer to Theorem 2 in the appendix where we comment on an alternative to Krasovskiĭ’s proof. Noting that

(11)  | inv(x) − inv(y)| = \frac{|x − y|}{|x| · |y|}

and putting (9), (10) and (11) together yields

|∂_{y}^{α}G_{ω^{c}}(x, y)| ≤ C|y|^{−|α|} ∑_{r≤|α|} |x|^{r}|x − y|^{2k−n−r}.

This proves the claim for ε = 1, while for arbitrary ε > 0 it follows from the previous reasoning and the observation that G_{(εω)^{c}}(x, y) := ε^{2k−n}G_{ω^{c}}(x/ε, y/ε). □

Step 2. Control outside a small annulus.

Given δ ∈ (0, \frac{d(0,∂Ω)}{3}), we define η_{δ} ∈ C_{c}^{∞}(Ω) such that η_{δ}(x) = 1 for all x ∈ B_{δ}(0)

and η_{δ}(x) = 0 for all x ∈ Ω \ B_{2δ}(0). Given ε ∈ (0, \frac{δ}{2 diam(ω)}) and x, y ∈ Ω_{ε}, we define

(12)  G̃_{ε,δ}(x, y) := η_{δ}(y)G_{(εω)^{c}}(x, y) + (1 − η_{δ}(y))G_{Ω}(x, y).

We get that

(−Δ)^{k}G̃_{ε,δ}(x, ·) = η_{δ}(−Δ)^{k}G_{(εω)^{c}}(x, ·) + (1 − η_{δ})(−Δ)^{k}G_{Ω}(x, ·)

+ ∑_{i<2k} (A_{i}(∇^{2k−i}η_{δ}, ∇^{i}G_{(εω)^{c}}(x, ·)) + A_{i}(∇^{2k−i}(1 − η_{δ}), ∇^{i}G_{Ω}(x, ·))) ,

where the A′_{i}s are contractions of suitable tensors, that is bilinear forms with smooth coefficients. Therefore, for any x ∈ Ω_{ε}, there exists f_{ε,δ,x} such that

(−Δ)^{k}G̃_{ε,δ}(x, ·) = δ_{x} + f_{ε,δ,x} in D′(Ω_{ε}).

Moreover, the pointwise control (7) yields

|f_{ε,δ,x}(y)| ≤ C · 1_{B_{2δ}(0)\B_{δ}(0)}|x − y|^{1−n}

for all x, y ∈ Ω_{ε}, x ≠ y. In particular, there exists C(δ) > 0 such that

(13)  ∥f_{ε,δ,x}∥_{L^{∞}(Ω_{ε})} ≤ C(δ) for all ε > 0 and x ∈ Ω_{ε,δ}

where

Ω_{ε,δ} := (Ω_{ε} ∩ B_{δ/2}(0)) ∪ (Ω_{ε} \ \overline{B_{3δ}(0)}) = Ω_{ε} \ (\overline{B_{3δ}(0)} \ B_{δ/2}(0)) .

Then it follows from elliptic theory that for any x ∈ Ω_{ε,δ}, there exists u_{x,ε,δ} ∈ W_{0}^{k,2}(Ω_{ε}) such that

(14)  \{ \begin{matrix}(−Δ)^{k}u_{x,ε,δ} = f_{ε,δ,x} & in Ω_{ε}, \\ ∂_{ν}^{(i)}u_{x,ε,δ} = 0 & for all i = 0, …, k − 1 on ∂Ω_{ε}. \end{matrix}

We claim that u_{x,ε,δ} ∈ C^{2k−1}(Ω_{ε}) for all ε, δ > 0 and x ∈ Ω_{ε,δ}. Moreover, there exists C(δ) > 0 such that

(15)  ∥u_{x,ε,δ}∥_{C^{2k−1}(Ω\B_{δ/4}(0))} ≤ C(δ)

for all admissible ε, δ > 0 and x ∈ Ω_{ε,δ}.

We prove this claim. For simplicity, we define

((−Δ)^{k/2}ψ)^{2} := \{ \begin{matrix}((−Δ)^{l}ψ)^{2} & if k = 2l is even \\ |∇(−Δ)^{l}ψ|^{2} & if k = 2l + 1 is odd. \end{matrix}

As a consequence, u ↦ ∥(−Δ)^{k/2}u∥_{2} is a norm on W_{0}^{k,2}(Ω_{ε}), the completion of C_{c}^{∞}(Ω_{ε}) for the usual norm. Multiplying (14) by u_{x,ε,δ} and integrating by parts yields with Hölder’s inequality

∫_{Ω}((−Δ)^{k/2}u_{x,ε,δ})^{2} dx = ∫_{Ω_{ε}}((−Δ)^{k/2}u_{x,ε,δ})^{2} dx
= ∫_{Ω_{ε}} f_{ε,δ,x}u_{x,ε,δ} dy ≤ ∥f_{ε,δ,x}∥_{\frac{2n}{n+2k}} ∥u_{x,ε,δ}∥_{\frac{2n}{n−2k}} .

Sobolev’s inequality yields the existence of C_{n,k} > 0 such that

∥u∥_{\frac{2n}{n−2k}} ≤ C_{n,k}∥(−Δ)^{k/2}u∥_{2}

for all u ∈ C_{c}^{∞}(ℝ^{n}). The density of C_{c}^{∞}(Ω_{ε}) in W_{0}^{k,2}(Ω_{ε}) allows to conclude that

∥u_{x,ε,δ}∥2_{\frac{2n}{n−2k}} ≤ C_{n,k}^{2}∥f_{ε,δ,x}∥_{\frac{2n}{n+2k}} ∥u_{x,ε,δ}∥_{\frac{2n}{n−2k}}

for all ε > 0 and x ∈ Ω_{ε,δ}. Therefore ∥u_{x,ε,δ}∥_{\frac{2n}{n−2k}} ≤ C′(δ).

It follows from elliptic theory (see for instance Agmon-Douglis-Nirenberg [1]) that for all p > 1 and all δ′ > 0, there exists C(δ′) > 0 such that

∥u_{x,ε,δ}∥_{W^{2k,p}(Ω\B_{δ′} (0))} ≤ C(δ′, p, Ω)(∥f_{ε,δ,x}∥_{p} + ∥u_{x,ε,δ}∥_{p}).

The claim (15) follows from this inequality, Sobolev’s inequalities and iterations.

It remains to gain control of u_{x,ε,δ} in B_{δ′}(0) \ (εω). To this end we consider η_{δ}u_{x,ε,δ} and observe that this function solves a Dirichlet problem in the exterior domain (\overline{εω})^{c}. Indeed, we have that

(−Δ)^{k}(η_{δ}u_{x,ε,δ}) = η_{δ}(−Δ)^{k}u_{x,ε,δ} + ∑_{i<2k} A_{i}(∇^{2k−i}η_{δ}, ∇^{i}u_{x,ε,δ})
= η_{δ}f_{ε,δ,x} + ∑_{i<2k} A_{i}(∇^{2k−i}η_{δ}, ∇^{i}u_{x,ε,δ}) := f̃_{ε,δ,x}

where the A_{i}’s are as above. We observe that

supp f̃_{ε,δ,x} ⊂ B_{2δ}(0) and ∥f̃_{ε,δ,x}∥_{∞} ≤ C(δ)

for all x ∈ Ω_{ε,δ}. Since η_{δ}u_{x,ε,δ} has compact support in ℝ^{n} \ εω and vanishes up to (k − 1)^{th} order on ∂(εω), Green’s representation formula (6) yields

(η_{δ}u_{x,ε,δ})(z) = ∫_{ℝ^{n}\εω} G_{(εω)^{c}}(z, y)f̃_{ε,δ,x}(y) dy

for all z ∈ ℝ^{n} \ \overline{εω}. Consequently, for any z ∈ B_{δ}(0) \ (εω), one gets

|u_{x,ε,δ}(z)| = |(η_{δ}u_{x,ε,δ})(z)| ≤ ∫_{ℝ^{n}\εω} |G_{(εω)^{c}}(z, y)f̃_{ε,δ,x}(y)| dy
≤ C(δ) ∫_{B_{2δ(0)}\εω} |y − z|^{2k−n} dz ≤ C(δ).

This inequality combined with (15) yields

(16)  ∥u_{x,ε,δ}∥_{L^{∞}(Ω_{ε})} ≤ C(δ) for all x ∈ Ω_{ε,δ}.

As a consequence, we find that

(−Δ)^{k}(G̃_{ε,δ}(x, ·) − u_{x,ε,δ}) = δ_{x} weakly in D′(Ω_{ε})

and ∂_{ν}^{(i)}(G̃_{ε,δ}(x, ·) − u_{x,ε,δ}) = 0 on ∂Ω_{ε} for all x ∈ Ω_{ε,δ} and all i = 0, …, k − 1. The uniqueness of the Green function implies that

(17)  G_{Ω_{ε}}(x, ·) = G̃_{ε,δ}(x, ·) − u_{x,ε,δ}

and then, using (6) and (16), we arrive at

(18) |G_{Ω_{ε}}(x, y)| ≤ |G̃_{ε,δ}(x, y)| + |u_{x,ε,δ}(y)| ≤ C(Ω, ω)|x − y|^{2k−n} + C(δ)
(19)  ≤ C(δ)|x − y|^{2k−n}

for all x ∈ Ω_{ε,δ} = (Ω_{ε} ∩ B_{δ/2}) ∪ (Ω_{ε} \ \overline{B}_{3δ}(0)) and all y ∈ Ω_{ε}.

Step 3.

Conclusion of the proof of Theorem 1. We fix δ_{0} ∈ (0, \frac{d(x_{0},∂Ω)}{21}). We apply Step 2 with δ := δ_{0} and to δ := 7δ_{0}. Since Ω_{ε} = Ω_{ε,δ_{0}} ∪ Ω_{ε,3δ_{0}}, it follows from (18) that there exists C > 0 such that

|G_{Ω\\overline{εω}}(x, y)| ≤ C|x − y|^{2k−n} for all x, y ∈ Ω \ \overline{εω}, x ≠ y.

This proves Theorem 1 for q = 1/42. For q ∈ (1/42, 1), instead of δ/2, δ, 2δ, 3δ, in Step 2 one has to work with δ/(1 + σ), δ, (1 + σ)δ, (1 + 2σ)δ with σ > 0 sufficiently close to 0. Alternatively one may argue that for

ε ∈ [(1/42)d(x_{0}, ∂Ω)/ diam(ω), qd(x_{0}, ∂Ω)/ diam(ω))

the boundaries of the Ω_{ε} enjoy uniform C^{2k,θ}-properties so that (3) holds uniformly with respect to these ε. □

Step 4.

Proof of Proposition 1. We argue by contradiction and assume that there exist 1 ≤ r ≤ 2k and C > 0 such that

(20)  |x − y|^{n−2k+r}|∇_{y}^{r}G_{Ω_{ε}}(x, y)| ≤ C

for all x, y ∈ Ω_{ε}, x ≠ y, uniformly in ε → 0. For any x, y ∈ (ε^{−1}Ω) \ ω, we define G_{ε}(x, y) := ε^{n−2k}G_{Ω_{ε}}(εx, εy). It follows from (12), (15), and (17) that for any x ∈ ω^{c}, we have that

lim_{ε→0} G_{ε}(x, y) = G_{ω^{c}}(x, y)

in C_{loc}^{0}(ℝ^{n} \ (ω ∪ {x})). Since (−Δ)^{k}G_{ε}(x, ·) = 0 and G_{ε}(x, ·) vanishes on ∂ω up to order (k − 1), elliptic regularity yields convergence in C_{loc}^{2k}(ω^{c} \ {x}). Rewriting (20) for G_{ε} and passing to the limit ε → 0 yields

(21)  |x − y|^{n−2k+r}|∇_{y}^{r}G_{ω^{c}}(x, y)| ≤ C

for all x, y ∈ ℝ^{n} \ ω, x ≠ y. We fix x ≠ 0 and we define G_{R}(z) := R^{n−2k}G_{ω^{c}}(Rx, z) for all z ∈ ω^{c} and R > R_{0} large enough. It follows from the explicit expression of G_{ω^{c}} in (5) that

(22)  lim_{R→+∞} G_{R}(z) = G(z) := |x|^{2k−n}|z|^{2k−n}G_{ω_{0}} (0, \frac{z}{|z|^{2}} )

in C_{loc}^{0}(ℝ^{n} \ ω). Since (−Δ)^{k}G_{R} = 0 and G_{R} vanishes on ∂ω up to order (k − 1), elliptic regularity yields the convergence of (G_{R}) to G in C_{loc}^{2k}(ω^{c}). On the other hand, (21) may be rewritten as

|∇^{r}G_{R}(z)| ≤ CR^{−r}|x − R^{−1}z|^{2k−n−r}

for z in a compact sudomain of ℝ^{n} \ ω and R large enough. Since r ≥ 1, passing to the limit R → +∞ yields ∇^{r}G = 0 in ℝ^{n} \ ω, which contradicts the explicit expression (22) of G. This concludes the proof of Proposition 1. □

Step 5.

Proof of Proposition 2. Assume that (ii) does not hold. Then there exists a C^{2k,θ}- smooth bouned domain ω_{0} ⊂ ℝ^{n} such that G_{ω_{0}} attains some negative values, say at (x_{0}, y_{0}) ∈ ω_{0} × ω_{0}, x ≠ 0. We define ω := (inv(ℝ^{n} \ \overline{ω_{0}})) ∪ \{0\} and Ω_{ε} := Ω \ \overline{εω} where ε > 0 is small and Ω is a smooth bounded domain containing 0. It follows from (12), (16), and (17) that

lim_{ε→0} ε^{n−2k}G_{Ω_{ε}}(εx, εy) = G_{ω^{c}}(x, y) = |x|^{2k−n}|y|^{2k−n}G_{ω_{0}}(inv(x), inv(y))

for all x, y ∈ ℝ^{n} \ \overline{ω}. Choosing x := inv(x_{0}) and y := inv(y_{0}) yields

lim_{ε→0} G_{Ω_{ε}}(εx, εy) = −∞,

and then (i) does not hold. Conversely, if (ii) holds, then (i) holds. □

Appendix A. Pointwise control of the Green function for fixed domains

The following result, under stronger smoothness assumptions on Ω but at the same time in a more general context, is due to Krasovskiĭ [8, 9]:

Theorem 2. Let Ω ⊂ ℝ^{n} be a C^{2k,θ}-smooth bounded domain of ℝ^{n} with 2k < n, θ ∈ (0, 1), and k ≥ 1. Let G_{Ω} be the Green function for (−Δ)^{k} with Dirichlet boundary condition. Then for all 0 ≤ r ≤ 2k, there exists C(Ω, r) > 0 such that

(23)  |∇_{y}^{r}G_{Ω}(x, y)| ≤ C|x − y|^{2k−n−r}

for all x, y ∈ Ω, x ≠ y.

We sketch here an alternative proof.

Proof. The case r = 0 and k = 2 under the smoothness assumptions as in the theorem is treated in Grunau-Robert [5, Theorem 4] (see also Gazzola-Grunau-Sweers [3, Propositions 4.22 and 4.23] for an exposition in book form). By making the obvious changes one may check that the proof can be extended to any k ≥ 1 and n > 2k. (Only the discussion of the smaller dimensions n ≤ 2k requires more care.) This means that there exists a constant C(Ω) > 0 such that

(24)  |G_{Ω}(x, y)| ≤ C(Ω)|x − y|^{2k−n}

for all x, y ∈ Ω, x ≠ y. We fix r ≥ 1 and we prove (23) by using local elliptic estimates and rescaling arguments. We proceed as in Grunau-Gazzola-Sweers [3, Prop. 4.23] and use the following local Schauder estimate from Agmon-Douglis-Nirenberg [1, Theorem 9.3] which holds true also close to ∂Ω. For any two concentric balls B_{R} ⊂ B_{2R} and any polyharmonic function v on B_{2R} ∩ Ω satisfying homogeneous Dirichlet boundary conditions on B_{2R} ∩ ∂Ω we have

(25)  ∥∇^{r}v∥_{L^{∞}(B_{R}∩Ω)} ≤ \frac{C}{R^{r}} ∥v∥_{L^{∞}(B_{2R}∩Ω)}.

The constant is uniform in R; the behaviour with respect to (small) R is obtained by means of scaling.

Keeping x ∈ Ω fixed, for any y ∈ Ω \ {x} we choose R = |x − y|/4 and apply (25) and (24) in B_{R}(y) ⊂ B_{2R}(y) to G_{Ω}(x, ·). This proves (23).

□

References

[1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727.

[2] T. Boggio, Sulle funzioni di Green d’ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97–135.

[3] F. Gazzola, H.-Ch. Grunau, and G. Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin etc., 2010.

[4] C. R. Graham, R. Jenne, L. J. Mason, and G. A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), no. 3, 557–565.

[5] H.-Ch. Grunau and F. Robert, Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions, Arch. Ration. Mech. Anal. 195 (2010), no. 3, 865–898.

[6] H.-Ch. Grunau, F. Robert, and G. Sweers, Optimal estimates from below for biharmonic Green functions, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2151–2161.

[7] J. Hadamard, Sur certains cas intéressants du problème biharmonique, Œuvres de Jacques Hadamard, Tome III, CNRS Paris, 1968, pp. 1297-1299. Reprint of: Atti IV Congr. Intern. Mat. Rome 12-14, 1908.

[8] Ju. P. Krasovskiĭ, Investigation of potentials connected with boundary value problems for elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 587–640 (Russian); English transl., Math. USSR Izv. 1 (1967), 569–622.

[9] , Isolation of the singularity of the Green’s function, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 977–1010 (Russian); English transl., Math. USSR Izv. 1 (1967), 935–966.

[10] Ch. Loewner, On generation of solutions of the biharmonic equation in the plane by conformal mappings, Pacific J. Math. 3 (1953), 417–436.

[11] M. Nakai and L. Sario, Green’s function of the clamped punctured disk, J. Austral. Math. Soc., Ser. B 20 (1977), 175–181.

[12] M. Nicolesco, Les fonctions polyharmoniques, Hermann, Paris, 1936.

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

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

E-mail address: Frederic.Robert@univ-lorraine

Institut Élie Cartan, Université de Lorraine, B.P. 70239, 54506 Vandœuvre-lès-Nancy Cedex, France