Multidimensional Kolmogorov-Petrovsky Test for the Boundary Regularity and Irregularity of Solutions to the Heat Equation

where f∗, u∗ (or f ∗, u∗) are lower (or upper) limit functions of f and u, respectively. Assume that u is the generalized solution of the FBVP constructed by Perron’s supersolutions or subsolutions method (see [1, 6]). It is well known that, in general, the generalized solution does not satisfy (1.3). We say that a point (x0, t0) ∈ ∂Ωδ is regular if, for any bounded function f : ∂Ω→R, the generalized solution of the FBVP constructed by Perron’s method satisfies (1.3) at the point (x0, t0). If (1.3) is violated for some f , then (x0, t0) is called irregular point.