This article has appeared in Mathematische Nachrichten, 179, 89-102 (1996). Copyright: Mathematische Nachrichten, Wiley-Verlag.

Positivity for perturbations

of polyharmonic operators with Dirichlet boundary conditions in two dimensions.

Hans-Christoph Grunau and Guido Sweers

Abstract

Higher order elliptic partial differential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shown that in ℝ^{2} small perturbations of polyharmonic operators and of the domain preserve the maximum principle. Hence the Green function for the clamped plate equation on an ellipse with small eccentricity is positive.

Keywords: maximum principle, polyharmonic, higher order elliptic, Green function, clamped plate equation

1 Historical comments and the main result

Maximum and comparison principles have proved to be a powerful tool in the theory of second order elliptic differential equations. So, for a better understanding of higher order elliptic differential equations it is an obvious step to investigate the question whether similar results do exist there. That is, if Ω is an appropriate bounded domain, does a function u that satisfies the higher order elliptic differential inequality Lu ≥ 0, with zero Dirichlet boundary condition, have a fixed positive sign? As an example one may think of the clamped plate equation

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

Here the question can be rephrased as:

For which shapes does upwards pushing imply upwards bending?

The pushing resp. bending is denoted by f, resp. u and Ω ⊂ ℝ^{2} is the shape.

Throughout this paper only two dimensional domains Ω ⊂ ℝ^{2} will be considered, if nothing different is stated. We have to leave the question open, whether the results of the present paper can be generalized to higher dimensions.

Boggio [3] (1901) and Hadamard [13] (1908) conjectured that in arbitrary reasonable domains Ω, f ≥ 0 implies u ≥ 0. Boggio in [4] could show this in case of the ball (Ω = B = unit ball). Furthermore, for any polyharmonic operator, (−Δ)^{m} , and balls B ⊂ ℝ^{n}, he calculated the Green function G_{m,n} (·, ·) for the Dirichlet problem and showed that G_{m,n} (·, ·) > 0 in B^{2}. In 1909 Hadamard [14] already knew, that the positivity conjecture is false in annuli with small inner radius. But he, and also Boggio, as Hadamard mentioned, believed, that there is no serious doubt, that Green’s function should be positive at least in convex domains.

Starting about 40 years later, numerous counterexamples [5], [6], [8], [9], [16], [17], [18], [22], [24] disprove the Boggio-Hadamard conjecture. The most striking examples have been found by Coffman, Duffin and Garabedian. For example, Garabedian ([9], see also [10, p. 275]) found that in an ellipse in ℝ^{2}, with the ratio of the half axes ≃ 2, the Green function for the biharmonic operator Δ^{2} changes sign (for an elementary proof, see also [22]). Coffman and Duffin [6] could show the same result, i.e. change of sign of Green’s function, in rectangles, including the square. That means that neither in arbitrarily smooth, uniformly convex domains nor in rather symmetric domains, we may expect Green’s function to be positive. Things are still worse: even the first Dirichlet-eigenfunction of Δ^{2} need not be unique and of one sign, see [5], [16].

In a recent paper [12] we could show that a polyharmonic operator in a ball which is slightly perturbed in the lower order terms still has the positivity preserving property mentioned above. In the present paper we shall show that in two dimensions one can even allow some small perturbation in the highest order term of the operator as well as in the shape of the domain and still has the positivity preserving property.

The higher order elliptic problem that we consider is

\{ \begin{matrix}Lu & ≥ & 0 & in Ω, \\ D_{m}u & = & 0 & on ∂Ω, \end{matrix}  (1)

where

L = (− ∑_{1≤i≤j≤2} a_{ij} (x) \frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} )^{m}+ ∑_{|α|≤2m−1} b_{α} (x) (\frac{∂}{∂x_{1}} )^{α_{1}} (\frac{∂}{∂x_{2}} )^{α_{2}} , (2)

a_{ij} ∈ C^{2m−1,γ}(\overline{Ω}), b_{α} ∈ C^{0,γ}(\overline{Ω}) and D_{m}u = 0 on ∂Ω is the zero Dirichlet boundary condition:

(\frac{∂}{∂x})^{α} u_{|_{∂Ω}} = 0 for all α with |α| ≤ m − 1.

For the sake of easy statement we define closeness of domains and operators.

Definition 1.1 Let ε > 0. We call Ω ε-close in C^{k,γ}-sense to Ω^{∗}, if there exists a C^{k,γ}-mapping g : \overline{Ω^{∗}} → \overline{Ω} such that g (\overline{Ω^{∗}}) = \overline{Ω} and

∥g − Id∥_{C^{k,γ}(\overline{Ω^{∗}})} ≤ ε.

Remark 1. For convex Ω^{∗}, k ≥ 1 and ε small we easily find that g is bijective and even that g^{−1} ∈ C^{k,γ}(\overline{Ω}), ||g^{−1} − Id||_{C^{k,γ}(\overline{Ω})} = O(ε). Local injectivity follows directly. Since Ω^{∗} is assumed to be convex, we find

\begin{aligned}|g (x) − g (y)| & ≥ & |x − y| − |g (x) − x − (g (y) − y)| & ≥ \\ & ≥ & |x − y| − ∥(g − Id)′∥_{C^{0}} |x − y| & ≥ & (1 − ε) |x − y| , \end{aligned}

implying that g is globally injective.

Definition 1.2 Let ε > 0 and let L be as above. We call the operator L ε-close in C^{k,γ}-sense to (−Δ)^{m} on \overline{Ω}, if additionally a_{ij} ∈ C^{k,γ}(\overline{Ω}) and

\begin{matrix}∥a_{ij} − δ_{ij}∥_{C^{k,γ}(\overline{Ω})} ≤ ε, \\ ∥b_{α}∥_{C^{0}(\overline{Ω})} ≤ ε. \end{matrix}

Remark 2. For ε small, L is uniformly elliptic.

Our main result shows that it is the large deviation from the constant flexure of the ball that yields a change of sign for the Green function:

Theorem 1.3 There exists ε_{0} = ε_{0}(m) > 0 such that, for ε ∈ [0, ε_{0}) we have: If ∂Ω ∈ C^{2m,γ}, Ω is ε-close in C^{2m}-sense to B and L is ε-close in C^{2m−1,γ}-sense to (−Δ)^{m} on \overline{Ω}, then every u ≢ 0 that satisfies (1) is strictly positive in Ω.

Remark 3. In principle ε_{0} = ε_{0}(m) could be calculated explicitly. We expect ε_{0}(m) ↘ 0 as m → ∞.

The theorem will be proven in two steps. First we will assume a_{ij} = δ_{ij} and consider “small” perturbations of the domain, see Section 2. This result and the theory of canonical forms will allow for perturbations of the leading coefficients of the operator too, see Section 3.

Corollary 1.4 Let E_{ε} = {(x_{1}, x_{2}) ; x_{1}^{2} + (1 + ε) x_{2}^{2} ≤ 1} denote ellipses close to the unit ball. There exists ε_{m} > 0 such that for all |ε| ≤ ε_{m} the Green function is positive for

\{ \begin{matrix}(−Δ)^{m} u = f & in E_{ε}, \\ D_{m}u = 0 & on ∂E_{ε}. \end{matrix}

Remark 4. In 1951 Garabedian showed in [9] (see also [10, p. 275]) that the Green function on some eccentric ellipses (ratio of half axes ≈ 2 for m = 2) changes sign. Our Corollary answers the question whether the Green function changes sign on any ellipse that is not a ball.

2 Domain perturbations

Throughout the rest of this paper we may assume m > 1.

We take the crucial lemma from our previous paper [12, sect. 5], which has been proved by means of Green’s function estimates.

Lemma 2.1 There is a δ_{0} = δ_{0}(m) > 0 such that the following holds. Let Ω be a simply connected, bounded smooth C^{2m,γ}-domain, L be as in Theorem 1.3 with a_{ij} = δ_{ij}. Let h : B → Ω be a biholomorphic mapping with h ∈ C^{2m,γ}(\overline{B}), h^{−1} ∈ C^{2m,γ}(\overline{Ω}).

If

\begin{aligned}||h − Id||_{C^{2m−1}(\overline{B})} & ≤ & δ_{0}, \\ ||b_{α}||_{C^{0}(\overline{Ω})} & ≤ & δ_{0}, \end{aligned}

for |α| < 2m, then the Green function of L in Ω under homogeneous Dirichlet boundary conditions is positive.

Remark 1. The existence of holomorphic mappings with qualitative properties as in the lemma is ensured by theorems of Riemann and Warschawski, see [19].

Remark 2. An explicit conformal mapping from the unit ball to any ellipse is given in the note of H. A. Schwarz [21]. So one could think that Corollary 1.4 above could be proven by an explicit analysis of this mapping. But, since elliptic functions are involved, this seems to be at least very difficult.

This example shows the need for a notion of closeness of domains as in Definition 1.1. Such a condition may be checked more easily.

Theorem 1.3 with a_{ij} = δ_{ij} will be proven, if we can show that C^{2m}-closeness to the ball with respect to differentiable mappings implies also C^{2m−1}-closeness with respect to holomorphic mappings.

Proposition 2.2 Let δ_{0} be given. Then there is some ε_{0} = ε_{0}(δ_{0}, m) > 0, such that for ε ∈ [0, ε_{0}) we have the following. If the C^{2m,γ}-domain Ω is ε-close in C^{2m}-sense to B, then there is a biholomorphic mapping h : B → Ω, h ∈ C^{2m,γ}(\overline{B}), h^{−1} ∈ C^{2m,γ}(\overline{Ω}) with

||h − Id||_{C^{2m−1}(\overline{B})} ≤ δ_{0}.

Proof. Let g : \overline{B} → \overline{Ω} be a mapping according to Definition 1.1 such that ||g − Id||_{C^{2m}(\overline{B})} < ε. The number ε is assumed to be small enough. According to [7], cf. also [23, sect. 4.2], the holomorphic mapping h^{−1} : Ω → B, which has the desired qualitative properties, may be constructed in the following way. First set

w(x) := 2πG(x, 0).

Here G is the Green function for −Δ in Ω under homogeneous Dirichlet condition. Next define the conjugate harmonic function

w^{∗}(x) := ∫_{1/2}^{x} (−\frac{∂}{∂ξ_{2}} w (ξ) dξ_{1} + \frac{∂}{∂ξ_{1}} w (ξ) dξ_{2}) ,

where the integral is taken with respect to any curve from \frac{1}{2} to x in Ω \ \{0\}. w^{∗} is well defined up to multiples of 2π. One finds that h^{−1} is well defined by

h^{−1}(x) := exp (−w (x) − iw^{∗} (x)) for x ∈ \overline{Ω},

where ℝ^{2} and ℂ are identified. The function h^{−1} maps 0 onto 0 and the point \frac{1}{2} somewhere into the positive real half-axis. Moreover, for x ∈ ∂Ω we find that |h^{−1}(x)| = |exp (−iw^{∗} (x))| = 1 and hence h^{−1} (∂Ω) ⊂ ∂B. For x ∈ Ω\ {0} we have w (x) > 0 and hence |h^{−1}(x)| < 1 implying h^{−1} (Ω) ⊂ B.

The Green function G(x, 0) is defined by

G(x, 0) = −\frac{1}{2π} (log |x| − r(x)) , x ∈ \overline{Ω},

where

\{ \begin{matrix}Δr = & 0 & in Ω, \\ r (x) = & φ(x) := log |x| & for x ∈ ∂Ω. \end{matrix}

Since

h^{−1}(x) = x exp (−r (x) − ir^{∗} (x)) for x ∈ \overline{Ω},

again identifying ℝ^{2} and ℂ, one finds that

||r||_{C^{2m−1}(\overline{Ω})} = O(ε)  (3)

will imply ||h^{−1}−Id||_{C^{2m−1}(\overline{Ω})} = O(ε) and consequently ||h−Id||_{C^{2m−1}(\overline{B})} = O(ε). The estimate in (3) will follow from the extension of the boundary data φ_{|_{∂Ω}} to some φ̂ on \overline{Ω} with

||φ̂||_{C^{2m}(\overline{Ω})} = O(ε).  (4)

Indeed, the estimate for ||r||_{C^{0}(\overline{Ω})} is immediate by the maximum principle. Furthermore, by means of elliptic estimates for second order equations (see [1, Theorem 7.3], [11, chapt. 6.4]) we find ||r||_{C^{2m−1,γ}(\overline{Ω})} = O(ε). Note that due to the closeness of Ω to B in C^{2m}-sense, according to Definition 1.1, the constants in these estimates may be chosen independently of Ω.

It remains to show the existence of some φ̂ that satisfies (4). This is done as follows. Since Ω is ε-close to B in C^{2m}-sense, one can show that (φ ∘ g)_{|_{∂B}} may be extended to φ_{g} on \overline{B} with ||φ_{g}||_{C^{2m}(\overline{B})} = O(ε). That means we have to estimate the “tangential derivatives” of φ ∘ g_{|_{∂B}} only.

Set ψ (t) := φ (g (cos t, sin t)). We are done, if we have shown that

max_{j=0,...,2m} max_{t∈[0,2π]} |(\frac{d}{dt})^{j} ψ| = O(ε).

We observe that ψ(t) = O(ε), since log |g(cos t, sin t)| = log (1 + O(ε)) = O(ε). Let us denote g̃(t) := g(cos t, sin t). For j ≥ 1 a tedious application of chain and product rule yields:

(\frac{d}{dt})^{j}ψ = (\frac{d}{dt})^{j}(φ ∘ g̃) =

= ∑_{|α|=1}^{j} ((D^{α}φ) ∘ g̃) ( ∑_{\begin{matrix}p_{1} + … + p_{|α|} = j \\ 1 ≤ p_{l} \end{matrix}} d_{j,α,p⃗} ∏_{l=1}^{|α|} (\frac{d}{dt})^{p_{l}}g̃^{(β_{l})})

with some suitable coefficients d_{j,α,p⃗}, β_{l} = 1 for l = 1, …, α_{1} and β_{l} = 2 for l = α_{1} + 1, …, |α|. We want to compare this with the corresponding expression with g replaced by Id. Denote g̃_{0}(t) = Id ∘ (cos t, sin t).

(\frac{d}{dt})^{j}ψ = ∑_{|α|=1}^{j} ( ((D^{α}φ) ∘ g̃ − (D^{α}φ) ∘ g̃_{0}) + (D^{α}φ) ∘ g̃_{0}) ×

× ( ∑_{\begin{matrix}p_{1} + … + p_{|α|} = j \\ 1 ≤ p_{l} \end{matrix}} d_{j,α,p⃗} ∏_{l=1}^{|α|} ( ((\frac{d}{dt})^{p_{l}}g̃^{(β_{l})} − (\frac{d}{dt})^{p_{l}}g̃_{0}^{(β_{l})}) + (\frac{d}{dt})^{p_{l}}g̃_{0}^{(β_{l})})) .

Since φ (g̃_{0}(t)) = log |(cos t, sin t)| ≡ 0, all expressions containing g̃_{0} only (and not a difference), sum up to zero. In the remaining sum, every term contains at least one factor of the form

(D^{α}φ) ∘ g̃ − (D^{α}φ) ∘ g̃_{0} or (\frac{d}{dt})^{p_{l}} (g̃^{(β_{l})} − g̃_{0}^{(β_{l})}) .

For ε small, each of these factors is at most O(ε). The other factors remain uniformly bounded with respect to ε ∈ [0, ε_{0}), ε_{0} chosen appropriately. We come up with

max_{j=0,...,2m} max_{t∈[0,2π]} |(\frac{d}{dt})^{j}ψ| = O(ε),

as required. □

3 Operator perturbations

Denote

L_{0} = − ∑_{1≤i≤j≤2} a_{ij} (x) \frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} .  (5)

First we assume that a_{ij} ∈ C^{3}(\overline{Ω}), that Ω is convex and that ∂Ω ∈ C^{3}. In order to simplify the notation we will use in the major part of this section (x, y) instead of (x_{1}, x_{2}) and set a := a_{11}, b := \frac{1}{2}a_{12} and c := a_{22}.

Second order linear equations in two variables can be reduced to a canonical form. See e.g. [10, p. 66-68]. For equations of elliptic type it means that there exist quasiconformal transformations (x, y) ↦ (ξ, η) such that with v (x, y) = V (ξ, η) we have

(L_{0}v) (x, y) = A (ξ, η) ( − ΔV (ξ, η) − w (ξ, η) · ∇V (ξ, η)) .  (6)

For references about reduction to canonical form see e.g. [2], [10], [20]. Here we have to deal with geometrically very simple domains. In this case some global properties of the mapping (x, y) ↦ (ξ, η) can be shown rather directly. For the reader’s convenience we include these proofs in the following.

For the transformation that we use we fix the boundary condition of the second component η to equal the second component of the identity.

We find that η satisfies the second order elliptic boundary value problem

\{ \begin{matrix}Mη = 0 in Ω, \\ η = y on ∂Ω, \end{matrix}  (7)

with M defined by

Mϕ = \frac{∂}{∂x} \frac{aϕ_{x} + bϕ_{y}}{\sqrt{ac − b^{2}}} + \frac{∂}{∂y} \frac{bϕ_{x} + cϕ_{y}}{\sqrt{ac − b^{2}}} .  (8)

Lemma 3.1 Let Ω be as above. Then (7) has a unique solution η ∈ C^{2,γ}(\overline{Ω}) and ∇η ≠ 0 in \overline{Ω}. Moreover, every level line ℓ_{η,t}(\overline{Ω}), defined by

ℓ_{η,t}(\overline{Ω}) = \{(x, y) ∈ \overline{Ω}; η (x, y) = t\} ,

with min\{y; (x, y) ∈ \overline{Ω}\} < t < max\{y; (x, y) ∈ \overline{Ω}\}, consists of one C^{1}-arc connecting two boundary points.

Proof. Existence and regularity for η follows from standard elliptic theory. We have to show that ∇η ≠ 0. First we fix p̂ ∈ ∂Ω. If the tangential direction τ \binom{}{p̂} at p̂ is not parallel (1, 0) , then \frac{∂η}{∂τ} = \frac{∂y}{∂τ} = τ_{2} \binom{}{p̂} ≠ 0 and ∇η is not parallel with n \binom{}{p̂} , where n \binom{}{p̂} is the normal direction. If τ \binom{}{p̂} and (1, 0) are parallel, then, since Ω is convex either η ≥ η \binom{}{p̂} or η ≤ η \binom{}{p̂} on ∂Ω and not identical 0, the maximum principle and Hopf’s boundary point Lemma imply \frac{∂η}{∂y} \binom{}{p̂} > 0. It shows that ∇η ≠ 0 on ∂Ω and hence that the Brouwer degree deg(∇η, Ω, 0) is well defined. Moreover, we will show that the homotopy [0, 1] × ∂Ω → ℝ^{2}, (t, p) ↦ t(0, 1) + (1 − t)∇η(p) is admissible. This is obvious for the two points p with τ(p) and (1, 0) parallel. If τ(p) and (1, 0) are not parallel, t(0, 1) + (1 − t)∇η(p) = 0 for t ∈ [0, 1], p ∈ ∂Ω would give: t ≠ 1, η_{x}(p) = 0, η_{y}(p) = t/(t−1), hence 0 ≠ τ_{2}(p) = \frac{∂η}{∂τ} (p) = η_{x}(p)τ_{1}(p)+η_{y}(p)τ_{2}(p) = τ_{2}(p) · t/(t − 1), a contradiction. Consequently we have

deg (∇η, Ω, 0) = deg ((0, 1) , Ω, 0) = 0.

Now suppose that ∇η \binom{}{p̂} = 0 for some p̂ ∈ Ω. Then the Carleman-Hartman-Wintner Theorem (see [20] and the appendix) implies that p̂ is an isolated zero of ∇η and moreover, that the local degree of ∇η at p̂ satisfies deg (∇η, B_{ε} \binom{}{p̂} , 0) < 0 (for some ε > 0 small). By the additivity property of the degree we obtain a contradiction with deg (∇η, Ω, 0) = 0. Hence ∇η ≠ 0 in \overline{Ω}.

Since ∇η ≠ 0 in \overline{Ω} the level lines are C^{1}. Note that the maximum principle implies that every component of a level set \overline{Ω}_{η,t}^{+} = \{p ∈ \overline{Ω}; η (p) > t\} or \overline{Ω}_{η,t}^{−} = \{p ∈ \overline{Ω}; η (p) < t\} has to intersect the boundary. Since Ω is convex and because of the boundary condition for η the boundary level sets (∂Ω)_{η,t}^{+} and (∂Ω)_{η,t}^{−} are connected

and hence the level sets \overline{Ω}_{η,t}^{+} and \overline{Ω}_{η,t}^{−} are (simply) connected. The last claim of the Lemma follows from the fact that

ℓ_{η,t}(\overline{Ω}) = ∂(\overline{Ω}_{η,t}^{+}) ∩ ∂(\overline{Ω}_{η,t}^{−}).
□

The first component ξ is defined up to a constant by the Beltrami equations

\{ \begin{matrix}ξ_{x} = \frac{bη_{x}+cη_{y}}{\sqrt{ac−b^{2}}} in Ω, \\ ξ_{y} = −\frac{aη_{x}+bη_{y}}{\sqrt{ac−b^{2}}} in Ω. \end{matrix}  (9)

We fix ξ by ξ (0, 0) = 0. Since the differential equation in (7) implies that ξ_{xy} = ξ_{yx} and since Ω is simply connected, we find that ξ is well defined by (9). Denote

F (x, y) := (ξ (x, y) , η (x, y))  (10)

and Ω^{∗} = F (Ω) .

Lemma 3.2 Let Ω, η and ξ be as above. Then the transformation F : \overline{Ω} → \overline{Ω^{∗}} is bijective and its Jacobian satisfies

J_{F} > 0 on \overline{Ω}.

Proof. Since F is continuous up to the boundary F (\overline{Ω}) = \overline{F (Ω)}. For the bijectivity note that ∇η ≠ 0 in \overline{Ω} implies ∇ξ ≠ 0 in \overline{Ω} and J_{F} > 0 on \overline{Ω}. Indeed

J_{F} = det (∇ξ ∇η) = \frac{1}{\sqrt{ac − b^{2}}} (aη_{x}^{2} + 2bη_{x}η_{y} + cη_{y}^{2}) > 0  (11)

implies ∇ξ ≠ 0 in \overline{Ω}. The inequality in (11) also implies that ξ is strictly monotone on any level line ℓ_{η,t}(\overline{Ω}). Since we know from the previous lemma that every level line ℓ_{η,t}(\overline{Ω}) consists of one component, the strict monotonicity of ξ implies that the transformation (x, y) → (ξ, η) is globally bijective. Notice that (11) in itself would only imply that the transformation is locally bijective. □

It remains to show that this transformation is close to the identity whenever L_{0} is close to −Δ.

Lemma 3.3 Now suppose additionally that ∂Ω is C^{2m,γ}. Let A, w, ξ and η be as in (6). For all δ > 0 there is ε > 0 such that if L_{0} is ε-close in C^{2m−1,γ}-sense to −Δ on \overline{Ω}, then

∥(ξ (·) , η (·)) − Id∥_{C^{2m,γ}} ≤ δ,
∥A (ξ, η) − 1∥_{C^{2m−1,γ}} ≤ δ,
∥w (ξ, η)∥_{C^{2m−2,γ}} ≤ δ.

Proof. First we consider η. Note that ϕ (x, y) = η (x, y) − y satisfies

\{ \begin{matrix}Mϕ = −\frac{∂}{∂x} (\frac{b}{\sqrt{ac−b^{2}}} ) − \frac{∂}{∂y} (\frac{c}{\sqrt{ac−b^{2}}} ) in Ω, \\ ϕ = 0 on ∂Ω, \end{matrix}

with M as in (8). Since L_{0} is ε-close in C^{2m−1,γ}-sense to −Δ we find that

∥\frac{∂}{∂x} (\frac{b}{\sqrt{ac − b^{2}}} ) − \frac{∂}{∂y} (\frac{c}{\sqrt{ac − b^{2}}} )∥_{C^{2m−2,γ}} = O (ε)

Since ∂Ω is C^{2m,γ} it follows again by elliptic estimates for second order equations (see [1, Theorem 7.3], [11, chapt. 6.4]) that

∥ϕ∥_{C^{2m,γ}} = O (ε) .  (12)

Hence ∥η_{x}∥_{C^{2m−1,γ}} = O (ε) and ∥η_{y} − 1∥_{C^{2m−1,γ}} = O (ε). Moreover, using (9) we find that ψ (x, y) = ξ (x, y) − x satisfies

\begin{aligned}ψ_{x} & = & \frac{b}{\sqrt{ac−b^{2}}} η_{x} + (\frac{c}{\sqrt{ac−b^{2}}} − 1) η_{y} + (η_{y} − 1) , \\ ψ_{y} & = & −\frac{a}{\sqrt{ac−b^{2}}} η_{x} − \frac{b}{\sqrt{ac−b^{2}}} η_{y} \end{aligned}

Since

\begin{aligned}∥\frac{a}{\sqrt{ac−b^{2}}} − 1∥_{C^{2m−1,γ}} & = & O (ε) , \\ ∥\frac{b}{\sqrt{ac−b^{2}}} − 0∥_{C^{2m−1,γ}} & = & O (ε) , \\ ∥\frac{c}{\sqrt{ac−b^{2}}} − 1∥_{C^{2m−1,γ}} & = & O (ε) , \\ ∥∇η − (0, 1)∥_{C^{2m−1,γ}} & = & O (ε) , \end{aligned}

we find ∥∇ψ∥_{C^{2m−1,γ}} = O (ε) . From ψ (0, 0) = 0 it follows that

∥ψ∥_{C^{2m,γ}} = O (ε) .  (13)

From (12) and (13) we conclude that ∥(ξ (·) , η (·)) − Id∥_{C^{2m,γ}} = O (ε) .

A and w remain to be estimated. With [10] we calculate that

\begin{aligned}A & = & aξ_{x}^{2} + 2bξ_{x}ξ_{y} + cξ_{y}^{2}, \\ w & = & \frac{1}{A} ( \begin{matrix}aξ_{xx} + 2bξ_{xy} + cξ_{yy} \\ aη_{xx} + 2bη_{xy} + cη_{yy} \end{matrix} )^{T} . \end{aligned}

Since ∥a − 1∥_{C^{2m−1,γ}} , ∥b∥_{C^{2m−1,γ}} , ∥c − 1∥_{C^{2m−1,γ}} , ∥ξ_{x} − 1∥_{C^{2m−1,γ}} , ∥ξ_{y}∥_{C^{2m−1,γ}} and hence ∥ξ_{xx}∥_{C^{2m−2,γ}} , ∥ξ_{xy}∥_{C^{2m−2,γ}} , ∥ξ_{yy}∥_{C^{2m−2,γ}} are all of order O (ε) we finally have

\begin{aligned}∥A − 1∥_{C^{2m−1,γ}} & = & O (ε) , \\ ∥w∥_{C^{2m−2,γ}} & = & O (ε) . \end{aligned}

□

Now we switch back to the original notation: a_{11} = a, a_{12} = 2b, a_{22} = c, (x_{1}, x_{2}) instead of (x, y).

Corollary 3.4 Let L be as in (2) and let ∂Ω ∈ C^{2m,γ}. For all δ > 0 there is ε > 0 such that the following holds. Suppose that the operator L_{0} as in (5) is ε-close in C^{2m−1,γ}-sense to −Δ. Then there is F : Ω̄ → ℝ^{2} such that

1. F (Ω) is δ-close in C^{2m,γ}-sense to Ω,

2. if u satisfies

\{ \begin{matrix}Lu & ≥ & 0 & in Ω, \\ D_{m}u & = & 0 & on ∂Ω, \end{matrix}

then U (F (x)) := u (x) satisfies

\{ \begin{matrix}L̃U & ≥ & 0 & in F (Ω) , \\ D_{m}U & = & 0 & on ∂F (Ω) , \end{matrix}  (14)

with

L̃ = (−Δ)^{m} + ∑_{|α|≤2m−1} b̃_{α} (x) (\frac{∂}{∂x_{1}} )^{α_{1}} (\frac{∂}{∂x_{2}} )^{α_{2}} .  (15)

and

∑_{|α|≤2m−1} ∥b_{α} − b̃_{α} ∘ F ∥_{C^{0}} < δ.  (16)

Proof. The transformation F is defined above. The property in 1. follows from the previous Lemma. The results in 2. follow from tedious but straigthforward calculus.

Indeed we find

( ∑_{1≤i≤j≤2} −a_{ij} \frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} )^{m} U (F (x)) =

= ( ∑_{1≤i≤j≤2} −a_{ij} \frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} )^{m−1} A ( (−ΔU) ∘ F (x) − w · ( (∇U) ∘ F (x))) =

= A^{m} ( ( (−Δ)^{m} U) ∘ F (x)) + ∑_{|α|≤2m−1} g_{α} (·) ((\frac{∂}{∂x})^{α} U) ∘ F (x) .

The coefficients g_{α} have at least one derivative of A of order between 1 and 2m − 2 as a factor. Furthermore, in the g_{α} appear derivatives up to order 2m − 2 of w, a_{ij} and F. Since ∥(\frac{∂}{∂x})^{β} A∥_{C^{0}} = O (ε) for all β with 1 ≤ |β| ≤ 2m − 2 the claim in (16) follows with help of Lemma 3.3. □

Proof of Theorem 1.3. Combine the results of Lemma 2.1, Proposition 2.2 and Corollary 3.4. □

A Appendix; on a result by Carleman, Hartman and Wintner

The next theorem is a corollary of the Carleman-Hartman-Wintner Theorem (see [20]). Their result has no direct extension to higher dimensional domains. For the sake of completeness we include a proof.

Theorem A.1 Let Ω be a domain in ℝ^{2} and let

L = ∑_{1≤i≤j≤2} a_{ij} (.) \frac{∂^{2}}{∂x_{i}∂x_{j}} + ∑_{1≤j≤2} b_{j} (.) \frac{∂}{∂x_{j}}

be uniformly elliptic on \overline{Ω} with a_{ij}, b_{j} ∈ C^{2}(\overline{Ω}). Suppose that ϕ ∈ C^{2}(\overline{Ω}) satisfies Lϕ = 0 in Ω and is non-constant. If p̂ ∈ Ω is such that ∇ϕ \binom{}{p̂} = 0, then p̂ is an isolated singularity:

there exists r > 0 such that \overline{B_{r} \binom{}{p̂}} ⊂ Ω and ∇ϕ ≠ 0 on \overline{B_{r} \binom{}{p̂}}\ \left\{{ \atop p̂}\right\} ,

and moreover

deg (∇ϕ, B_{r} \binom{}{p̂} , 0) < 0.

Proof. From the uniform ellipticity of L it follows that there exist λ_{1}, λ_{2} > 0 and an orthogonal matrix Q, with det Q = 1, such that

Q^{−1} ( \begin{matrix}a_{11}(p̂) & \frac{1}{2}a_{12}(p̂) \\ \frac{1}{2}a_{12}(p̂) & a_{22}(p̂) \end{matrix} ) Q = ( \begin{matrix}λ_{1} & 0 \\ 0 & λ_{2} \end{matrix} ) .

With the transformation U : ℝ^{2} → ℝ^{2}, defined by

U (z_{1}, z_{2}) = (Q ( \begin{matrix}(λ_{1})^{\frac{1}{2}} & 0 \\ 0 & (λ_{2})^{\frac{1}{2}} \end{matrix} ) ( \begin{matrix}z_{1} \\ z_{2} \end{matrix} ) + ( p̂_{1} ; p̂_{2} ))^{T}

we find that φ (z) := ϕ (Uz) − ϕ \binom{}{p̂} satisfies a uniformly elliptic equation L̂φ = 0 on U^{inv}Ω with the operator L̂ satisfying â_{11} (0) = â_{22} (0) = 1, â_{12} (0) = 0. Moreover φ (0) = 0 and ∇φ (0) = 0. Hence we may use the Carleman-Hartman-Wintner Theorem (see [20, Th. 7.2.1]). We will also use the result [20, Th. 7.2.4]. Since φ (z) = O (|z|) as |z| → 0 it follows that either φ (z) ≡ 0 on U^{inv}Ω, or there exists m ∈ ℕ^{+} with

lim_{|z|→0} \frac{φ_{z_{1}} − iφ_{z_{2}}}{(z_{1} + iz_{2})^{m}} = α ∈ ℂ\ \{0\} .  (17)

If φ (z) ≡ 0 on U^{inv}Ω then ϕ(x) ≡ ϕ(p̂) on Ω. Now suppose that φ (z) ≢ 0. Then there is r^{∗} > 0 with \overline{B_{r^{∗}} (0)} ⊂ U^{inv}Ω and ∇φ (z) ≠ 0 for z ∈ \overline{B_{r^{∗}} (0)}\ \{0\}. Hence 0 is an isolated zero of ∇φ and a homotopy argument shows that

deg (∇φ, B_{r^{∗}} (0) , 0) =
= deg ((Re(α(z_{1} + iz_{2})^{m}) , −Im(α(z_{1} + iz_{2})^{m})) , B_{r^{∗}} (0) , 0) ,

implying that deg (∇φ, B_{r^{∗}} (0) , 0) = −m < 0. Now take a ball B_{r}(p̂) such that B_{r}(p̂) ⊂ UB_{r^{∗}} (0). Since ∇φ ≠ 0 on \overline{B_{r^{∗}} (0)}\ (U^{inv}B_{r}(p̂)) we have

deg (∇φ, B_{r^{∗}} (0) , 0) = deg (∇φ, U^{inv} (B_{r} \binom{}{p̂}) , 0) .

According to Heinz [15, Lemma 7], this degree may be defined as follows. For some

ε > 0 we have |∇φ| ≥ ε on ∂ (U^{inv} (B_{r} \binom{}{p̂})). Any χ ∈ C^{0}(ℝ^{2}, ℝ) with suppχ ⊂

B_{cε}(0), c ∈ (0, 1) to be chosen below, and ∫_{ℝ^{2}} χ(z) dz = 1 is a normalized admissible testing function for ∇φ. We have

−m = deg (∇φ, U^{inv} (B_{r} \binom{}{p̂}) , 0) =
= ∫_{U^{inv}(B_{r} \binom{}{p̂})} χ (∇φ (z)) det ( \begin{matrix}φ_{11} (z) & φ_{12} (z) \\ φ_{21} (z) & φ_{22} (z) \end{matrix} ) dz =

= ∫_{U^{inv}(B_{r} \binom{}{p̂})} χ (∇ϕ (Uz) U′) det ((U′)^{T} ( \begin{matrix}ϕ_{11} (Uz) & ϕ_{12} (Uz) \\ ϕ_{21} (Uz) & ϕ_{22} (Uz) \end{matrix} ) U′) dz =

= ∫_{B_{r} \binom{}{p̂}} χ (∇ (ϕ (p)) U′) det ( \begin{matrix}ϕ_{11} (p) & ϕ_{12} (p) \\ ϕ_{21} (p) & ϕ_{22} (p) \end{matrix} ) |det (U′)| dp.

Here

U′ = Q ( \begin{matrix}λ1^{\frac{1}{2}} & 0 \\ 0 & λ2^{\frac{1}{2}} \end{matrix} ) .

We note that det U′ > 0 and introduce

χ̃ ∈ C^{0}(ℝ^{2}, ℝ), χ̃(p) := det (U′) χ (pU′) .

As

∫_{ℝ^{2}} χ̃(p) dp = ∫_{ℝ^{2}} det (U′) χ (pU′) dp = ∫_{ℝ^{2}} χ(z) dz,

χ̃ is a normalized admissible testing function for ∇ϕ, provided that c above has been chosen sufficiently small. We conclude by

0 > −m = ∫_{B_{r}(p̂)} χ̃ (∇ϕ (p)) det ( \begin{matrix}ϕ_{11} (p) & ϕ_{12} (p) \\ ϕ_{21} (p) & ϕ_{22} (p) \end{matrix} ) dp = deg (∇ϕ, B_{r} \binom{}{p̂} , 0) .

□

Acknowledgements. The authors gratefully acknowledge support from the EU through the project Reaction - Diffusion Equations under contract erbchrxct 930409.

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