Smoothness of the Singularity for the Heat Equation in a Polygonal Domain (Part Two) PDF

Keywords

mathematics, singularity, polygons

Abstract

We continue our study of the heat equation with Cauchy- Dirichlet conditions in the non convex polygonal domain Ω, described by the time variable t and one space variable x. The second member f of the heat equation is in L²(Ω), the space of functions the squares of which are integrable in Ω. We look for the solution u in a non symmetric Sobolev space H^(r,2r) (Ω) defined as an interpolation space between H^(1,2) (Ω) and L²(Ω) where H^(1,2) (Ω) = {u ϵ L² (Ω): u'₁, u'ₓ, u"ₓ, ϵ L² (Ω)}. It is known (Sadallah 1976, 1983) that u is smooth (i.e. belongs to H^(1,2) (Ω)) when the domain Ω is convex, but in general, this does not hold in a non convex polygon. The First Part of this work (Sadallah 1989) was devoted to the special case in which Ω is a non convex domain and is the product of two rectangles. It had been proved there, that for all f in L²(Ω) there exist two functions v, w, such that u= v+w where v ϵ H^(1,2) (Ω) and w not an element of H^(1,2) (Ω), Our main result was: The singularity w ϵ H^(r,2r) (Ω) iff r <3/4. In this second part, we prove that the same result remains unchanged in the general case (i.e. when Ω is not necessary product of two rectangles). The proof uses the First Part of this work and some other results of the author (Sadallah 1976) as well as the interpolation theory.