Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order 2m≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2m\ge 4$$\end{document} have been shown so far only when the principal part of L is the polyharmonic operator (-Δ)m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^m$$\end{document}. In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.


Introduction.
In a series of works [3,4,6,7,9,13], Dall'Acqua, Meister, Pulst, Robert, Sweers, and the present author have studied the question whether (suitably modified) Green functions in bounded smooth domains Ω ⊂ R n under Dirichlet boundary conditions for elliptic operators L of higher order 2m ≥ 4 may be estimated from above and below by the same positive prototype function. Such kind of result may be considered as a kind of substitute or relaxation of the maximum principle which in its strong form is true only for second order operators. Employing perturbative and blow-up arguments, all these works are based on Boggio's explicit formula for the polyharmonic Green function under Dirichlet boundary conditions in balls (see [2, p. 126] and also [5,Lemma 2.27]) and similarly in half spaces (see [5,Remark 2.28]). This does not only show its positivity in these special domains but does also 96 H.-C. Grunau Arch. Math.
allow for precise estimates from above and below, see [5,Theorem 4.6] and Remark 2(a) below. In order to keep this note short and as simple as possible, I shall consider only the (simplest) generic case n > 2m.
The "small" dimensions n = 2, . . . , 2m require more effort, but related results will also hold there. The most general class of operators, which could be treated so far, was considered by Pulst [13]. He could show that in general bounded smooth domains the estimate (2) below holds for operators of the form where the coefficients obey symmetry, smoothness, and coercivity conditions.
The goal here is to show that the same estimates can be proved for a principal part with variable coefficients, as long as this stays in the class of powers of second order elliptic operators.
let Ω ⊂ R n be a bounded C ∞ -smooth domain and L be a uniformly elliptic symmetric operator of the form For the coefficients, we assume that and that there exist numbers Let G Ω,L denote the Green function in Ω for the following Dirichlet problem Then there exist constants c 1 ≥ 0, c 2 > 0, depending on the domain Ω and the elliptic operator L, such that we have the following Green function estimate: for all x, y ∈ Ω, where and d Ω (x) := dist(x, ∂Ω).

Remark 2. (a)
If L = (−Δ) m and Ω is a ball or a half space in R n , (2) holds true with c 1 = 0. This follows from Boggio's formula as already mentioned at the beginning of the introduction. Since n > 2m, one has a unique Green function even in the whole space and this is given by the positive fundamental solution G (−Δ) m ,R n (x, y) = c m,n |x−y| 2m−n with a suitable positive normalisation constant c m,n > 0. (b) The existence of a Green function is in the case of a principal part with nonconstant coefficients by no means obvious and is due to Krasovskiȋ [11,Theorem 3.3]. Thanks to the symmetry of the operator L, this is symmetric, i.e. G Ω,L (x, y) = G Ω,L (y, x). (c) Krasovskiȋ's work requires high regularity of the coefficients of L and of the domain. To keep the exposition as simple as possible, I just assume everything to be C ∞ -smooth. (d) The estimate from above follows from [4,10,11], see also [6], so that only the estimate from below has to be proved here. (e) With the techniques used here and developed in the mentioned works, one may admit also lower order terms as long as symmetry and coercivity assumptions are obeyed. (f) As it was investigated in [8], the situation becomes completely different when L is a general uniformly elliptic operator (even with constant coefficients), which cannot be written as a composition of second order operators, and when the dimension is large, i.e. n ≥ 2m + 2. Here, in general, even the fundamental solution is sign changing near its singularity. This means that close to its singularity the negative part of any corresponding Green function becomes unbounded of the same order of magnitude as the positive part. (g) For implications of two-sided estimates as in Theorem 1, one may see [14][15][16].
An interesting consequence of Theorem 1 is a uniform local positivity result, which was shown in the mentioned works before for the more special class of operators.

Local estimates from below.
Lemma 4. Suppose that the assumptions of Theorem 1 are satisfied. Then for each x 0 ∈ Ω, there exists a radius r = r x0 > 0 and a constant C = C x0 > 0 such that for all x, y ∈ Ω x0,r := Ω ∩ B r (x 0 ), x = y, one has where H Ω is defined in (3).
Proof. We assume by contradiction that no such positive numbers exist such that (5) is satisfied. This means that we may find sequences ( We shall rescale the domain and the Green function such that |x k −y k | becomes the new length unit. In the scaling limit, we shall end up with a Green function for a constant coefficient elliptic operator in the half space or in the whole space which is known to enjoy an estimate like (5) so that we end up with a contradiction. This key argument relies on knowing the Green functions in these prototype situations explicitly, see Remark 2(a) above.
To this end, we need to distinguish the cases whether the x k , y k are as close together as to the boundary or much closer.
Thanks to the triangle inequality This yields in view of (6) that d Ω (x k ), d Ω (y k ) → 0 and x 0 ∈ ∂Ω and due to our assumption (7) that Without loss of generality, we may assume that x 0 = 0 and that the first unit vector e 1 is the exterior unit normal to ∂Ω at x 0 .
For k large enough, we may definex k ∈ ∂Ω as the closest boundary point to x k . We introduce the rescaled Green functions These belong to the elliptic operators in Ω k . This can be seen as follows. For smooth v : Ω k → R, we define Sincex k → x 0 = 0, the exterior unit normal at ∂Ω converges to the first unit vector and so we conclude that In order to understand the limit of the G k , we observe first that Krasovskiȋ's estimates [11,Theorem 3.3] yield uniformly in k that Employing local elliptic estimates [1], we find a limit in L 1 loc and locally uniformly in H ×H \{(ξ, ξ) : ξ ∈ H}. This is symmetric, i.e. G(ξ, η) = G(η, ξ), obeys homogeneous Dirichlet boundary conditions on ∂H, and satisfies |G(ξ, η)| ≤ C|ξ − η| 2m−n .
We introduce the limit operator . One should observe that this operator can easily be transformed by means of a linear transformation into the polyharmonic (−Δ) m . To be more precise, let ..,n . We then introduce a diagonal dilation matrix C 2 with positive diagonal entries such that (C 1 · C 2 ) T · a ∞ ij i,j=1,...,n · (C 1 · C 2 ) becomes the n-dimensional identity matrix. Finally we introduce a further "rotation" C 3 ∈ SO(n) such that C := C 1 · C 2 · C 3 : H → H maps again the half space onto itself. Using this transformation, we find that where k m,n > 0 is a suitable constant and ξ * = (−ξ 1 , ξ 2 , . . . , ξ n ). So we conclude from [5,Theorem 4.6] that with a strictly positive constant c 4 . H H is as in (3) with d H (ξ) = |ξ 1 |. Since Boggio's formula has the same form in H as in the ball, (13) is deduced in H in the same way as in the ball. Moreover, as explained in [6,7], G H,L∞ is unique under the assumption In order to conclude that G = G H,L∞ , we only need to show that in the distributional sense. To this end, we consider any function ψ ∈ C ∞ 0 (H) and find from the fact that G k is the Green function for L k that one has for any ξ ∈ H and k large enough, The smoothness assumptions on the coefficients yield that a This shows that G is a Green function for L ∞ in H, which in view of its uniqueness explained above yields that Defining on the one hand, the assumption gives via (10) that On the other hand, we have After passing to a further subsequence, we find ξ, η ∈ H with ξ = lim k→∞ ξ k , η = lim k→∞ η k . In view of the local smooth convergence of G k to the Green function G H,L∞ , we see from (13) that there exists a positive constant c 5 > 0 such that for k large enough, This contradicts (15) and the proof of the lemma is complete in the first case.
Second case: The reasoning here is similar to the first case and I outline only the main steps. After passing to a subsequence, we may assume that This also implies that We obtain from the assumption (7) that in this case We rescale again such that |x k − y k | becomes the new length unit; however, here it is the x k which become the new origins.
These belong to the elliptic operators ij (ξ) := a ij (x k + |x k − y k |ξ) (17) in Ω k . In view of the assumptions in this case, we conclude that For the sequence G k , we find a limit in L 1 loc and locally uniformly in R n × R n \ {(ξ, ξ) : ξ ∈ R n }, which is symmetric and decays like (ξ, η) → |ξ − η| 2m−n at infinity. Thanks to the uniqueness of such kind of Green function, we conclude that in the sense just described we have where the constant coefficient operator L ∞ is obtained precisely as in the first part. I emphasise that here the assumption n > 2m simplifies the proof a lot because the uniqueness conclusion follows directly from Liouville's theorem for Since n > 2m by assumption and since d Ω (x)d Ω (y) is bounded on Ω, one finds a constant c > 0 such that c −1 2 H Ω (x, y) ≥ 2c 1 d Ω (x) m d Ω (y) m for |x − y| < c. The proof of Theorem 3 is complete.