Abstract

We study fundamental solutions of elliptic operators of order 2mge 4 with constant coefficients in large dimensions nge 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 nge 3, the polyharmonic operator (-Delta )^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 nge 2m+3 fundamental solutions of specific operators of order 2mge 4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to +infty and -infty 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.

Highlights

  • We focus our attention to uniformly elliptic operators of order 2m with constant coefficients which involve only the highest order derivatives, namely

  • 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 solution KL (x, . ) for any “pole” x ∈ Rn which is defined as a solution to the equations L∗ K L (x, . ) = δx and L K L ( . , x) = δx in the distributional sense where δx is the δ-distribution located at x

  • For given f ∈ C0∞(Rn), any fundamental solution yields a solution to the differential equation Lu = f in Rn by putting u(x) := K L (x, y) f (y) d y

Read more

Summary

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. Ai1,...,i2m ξi1 · · · ξi2m i1,...,i2m =1,...,n is called (possibly up to a sign) the symbol of the operator. Uniform ellipticity means that Q is strictly positive on the unit sphere, i.e.

Fundamental solutions
Green functions
Positivity questions
Previous results
Aim and results
Basic observations
Ellipticity and positive directions
A method of descent with respect to space dimension
Understanding “small” dimensions is sufficient
The first iteration

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.