ABSTRACT
First of all, we need to comment on the regularity of the vertex (0, 0) of the backward “fundamental” parabolae:
R(t) = l √−t, i.e., ϕ(τ) ≡ l = const. > 0. (141)
Then, problem (140) is considered on the fixed unbounded interval
Il = {−∞ < y < l}, (142)
so that the final conclusion entirely depends on spectral properties of B∗
in Il with Dirichlet boundary conditions. Since we need a sharp bound on the first eigenvalue, the clear conclusion on regularity/irregularity becomes rather involved, where numerics are necessary to fix final details. In addition, as we pointed out, in a more general setting for the fundamental backward paraboloids in IRN , the existence, uniqueness, and regularity of solutions in Sobolev spaces was first proved in a number of papers by Mihaˇilov in 1961-63 [297, 299, 300], and in [125], etc. Note that, in [299, p. 45], the zero boundary data were understood in the mean sense (i.e., in the L2-sense along a sequence of smooth internal contours, “converging” to the boundary).
