MULTIDIMENSIONAL KOLMOGOROV-PETROVSKY TEST FOR THE BOUNDARY REGULARITY AND IRREGULARITY OF SOLUTIONS TO THE HEAT EQUATION UGUR G. ABDULLA Received 25 August 2004 Dedicated to I. G. Petrovsky This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of RN+1 (N ≥ 2) for the diffusion (or heat) equation. The result implies asymptotic probability law for the standard Ndimensional Brownian motion. 1. Introduction and main result Consider the domain Ωδ = (x,t) ∈ RN+1 : |x| 0, N ≥ 2, x = (x1 ,.,xN ) ∈ RN , t ∈ R,. | MULTIDIMENSIONAL KOLMOGOROV-PETROVSKY TEST FOR THE BOUNDARY REGULARITY AND IRREGULARITY OF SOLUTIONS TO THE HEAT EQUATION UGUR G. ABDULLA Received 25 August 2004 Dedicated to I. G. Petrovsky This paper establishes necessary and sufficient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of RN 1 N 2 for the diffusion or heat equation. The result implies asymptotic probability law for the standard N-dimensional Brownian motion. 1. Introduction and main result Consider the domain Qs x t e RN 1 x h t -8 t 0 where s 0 N 2 x x1 . xN e RN t e R h e C -S 0 h 0 for t 0 and h t 1 0 as t 1 0. For u e ex 1 Qs we define the diffusion or heat operator N Du Ut - Au Ut - Uxx x t e Qs. i 1 A function u e ex t Qs is called parabolic in Qs if Du 0 for x t e Qs. Let f dQ R be a bounded function. First boundary value problem FBVP may be formulated as follows. Find a function u which is parabolic in Qs and satisfies the conditions f u u f for z e dQs 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 . We say that a point x0 t0 e dQs is regular if for any bounded function f dQ R the generalized solution of the FBVP constructed by Perron s method satisfies at the point x0 t0 . If is violated for some f then x0 t0 is called irregular point. Copyright 2005 Hindawi Publishing Corporation Boundary Value Problems 2005 2 2005 181-199 DOI 182 Multidimensional Kolmogorov-Petrovsky test The principal result of this paper is the characterization of the regularity and irregularity of the origin 0 in terms of the asymptotic behavior of h as t1 0. We write h t 2 tlogp t 1 2 and assume that p G C -g 0 p t 0 for -8 t 0 p t 1 0 as 11 0 and logp t o log t as 11 0 see Remark .
