Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equatio
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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| < h(t), −δ < t < 0 ,
(1.1)
where δ > 0, N ≥ 2, x = (x1 ,...,xN ) ∈ RN , t ∈ R, h ∈ C[−δ,0], h > 0 for t < 0 and h(t) ↓ 0 as t ↑ 0. 2,1 For u ∈ Cx,t (Ωδ ), we define the diffusion (or heat) operator Du = ut − ∆u = ut −
N
ux i x i ,
(x,t) ∈ Ωδ .
(1.2)
i=1 2,1 A function u ∈ Cx,t (Ωδ ) is called parabolic in Ωδ if Du = 0 for (x,t) ∈ Ωδ . Let f : ∂Ω → R be a bounded function. First boundary value problem (FBVP) may be formulated as follows. Find a function u which is parabolic in Ωδ and satisfies the conditions
f ∗ ≤ u∗ ≤ u∗ ≤ f ∗
for z ∈ ∂Ωδ ,
(1.3)
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. Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 181–199 DOI: 10.1155/BVP.2005.181
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Multidimensional Kolmogorov-Petrovsky test
The principal result of this paper is the characterization of the regularity and irregularity of the origin (ᏻ) in terms of the asymptotic behavior of h as t ↑ 0. We write h(t) = 2(t log ρ(t))1/2 , and assume that ρ ∈ C[−δ,0], ρ(t) > 0 for −δ ≤ t < 0; ρ(t) ↓ 0 as t ↑ 0 and
log ρ(t) = o log |t |
as t ↑ 0
(1.4)
(see Remark 1.2 concerning this condition). The main result of this paper reads as follows. Theorem 1.1. The origin (ᏻ) is regular or irregular according as 0−
N/2
ρ(t) logρ(t) t
dt
(1.5)
diverges or converges. For example, (1.5) diverges for each of the following functions
−1
ρ(t) = log |t | ,
−1 ρ(t) = log |t | log(N+2)/2 log |t| ,
−1 n (N+2)/2 ρ(t) = log |t | log log |t | logk |t | ,
n = 3,4,...,
(1.6)
k=3
where we use the following notation:
log2 |t | = log log |t |,
logn |t | = log logn−1 |t |,
n ≥ 3.
(1.7)
From another side, (1.5) converges for each function
−(1+)
ρ(t) = log |t |
,
− 1 ρ(t) = log |t | log(N+2)/2+ log |t | ,
−1 ρ(t) = log |t | log(N+2)/2 log |t | log1+ , 3 |t |
(1.8)
− 1 ρ(t) = log |t | log(N+2)/2 log |t | log3 |t | log1+ , 4 |t |
and so forth, where > 0 is sufficiently small number. If we take N = 1, then Theorem 1.1 coincides with the result of Petrovsky’s celebrated paper [6]. From
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