Differences between fundamental solutions of general higher order elliptic operators and of products of second order ope
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Mathematische Annalen
Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators Hans-Christoph Grunau1
· Giulio Romani2
· Guido Sweers3
Received: 12 February 2019 / Revised: 6 May 2020 © The Author(s) 2020
Abstract We study fundamental solutions of elliptic operators of order 2m ≥ 4 with constant coefficients in large dimensions n ≥ 2m, where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as n ≥ 3, the polyharmonic operator (−)m may no longer serve as a prototype for the general elliptic operator. It is known from examples of Maz’ya and Nazarov (Math. Notes 39:14–16, 1986; Transl. of Mat. Zametki 39, 24–28, 1986) and Davies (J Differ Equ 135:83–102, 1997) that in dimensions n ≥ 2m + 3 fundamental solutions of specific operators of order 2m ≥ 4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to +∞ and −∞ respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a “positive” direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for n = 2m, n = 2m + 2 and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order 2m in dimension n = 2m +2. On the other hand we show that in the dimensions n = 2m and n = 2m + 1 the fundamental solution of any such elliptic operator is always positive around its singularity.
Communicated by Y. Giga. Extended author information available on the last page of the article
123
H.-C. Grunau et al.
1 Introduction and main results 1.1 General constant coefficients elliptic operators We focus our attention to uniformly elliptic operators of order 2m with constant coefficients which involve only the highest order derivatives, namely L = (−1)m Q
∂ ∂ ,..., ∂ x1 ∂ xn
= (−1)m
Ai1 ,...,i2m
i 1 ,...,i 2m =1,...,n
∂ ∂ ··· , ∂ xi1 ∂ xi2m (1)
where the 2m-homogeneous characteristic polynomial Q(ξ ) =
Ai1 ,...,i2m ξi1 · · · ξi2m
i 1 ,...,i 2m =1,...,n
is called (possibly up to a sign) the symbol of the operator. Uniform ellipticity means then that Q is strictly positive on the unit sphere, i.e. there exists a constant λ > 0 such that ∀ξ ∈ Rn :
Q(ξ ) ≥ λ|ξ |2m .
1.2 Fundamental solutions In order to construct and to understand solutions u to the differential equation Lu = f for a given right-hand side f , one introduces the concept of a fundamental solu
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