Unfinished business: some open questions in classical analysis

Speaker: D. Armitage (QUB)

Time: 11:00AM
Date: Mon 3rd September 2007

Location: ENG226


Some easily stated open problems will be discussed.

(i) It is known (Zalcman, 1982) that the Radon transform is not injective: there exist non-trivial continuous functions f: R2 → R with zero (proper) integral on every (doubly infinite, straight) line. All known examples of such functions have extremely rapid overall growth. Can such a function have slow growth, or even be bounded? Is it true that a continuous function on 3 with zero integral on every line must be identically zero?

(ii) Every polygonal domain D in R2 has the Pompeiu property (PP): if f: R2 → R is continuous and ∫ σ(D) f(x)dx = 0 for every rigid motion σ, then f ≡ 0. The corresponding assertion for functions on the sphere S2 is false: there are infinitely many (non-congruent) regular spherical polygons that lack PP, and they can be characterised. But it still seems unclear whether, for example, all non-trivial regular spherical triangles have PP, and whether (up to congruence) the known example of a spherical square lacking PP is unique.

(iii) One formulation of the maximum principle asserts that if h is a non-constant harmonic function on a ball centred at the origin O in Rn and h(O) = 0, then h takes positive values and negative values on every neighbourhood of O. In the case n = 2, this can be quantified: it is easy to show that, with h as above, the subset of {x:||x|| < r} where h > 0 and the subset where h < 0 have roughly the same area. (The ratio of the areas tends to 1 as r → 0+.) What can be said in the case n ≥ 3?

(This talk is part of the IMS September Meeting 2007 series.)