TY - JOUR
T1 - The oblique derivative problem for the heat equation in lipschitz cylinders
AU - Brown, Russell M.
N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 1989/9
Y1 - 1989/9
N2 - We consider a class of initial-boundary value problems for the heatequation on (0, T) x Ω with Ω a bounded Lipschitz domain in Rn. On thelateral boundary, (0, T) x Ω = ∑T, we specify (α, ∇v) where ∇v denotesthe spatial gradient of the solution and α: ∑T→(x: ∣x∣u = 1) is a continuousvector field satisfying (α, v) ≥ p > 0 with v the unit normal to Ω. On theinitial surface, (0) xΩ, we require that the solution vanish. The lateral datais taken from Lp∑T(I.t) ¦ For p ϵ (2 - ϵ, ∞), we show existence and uniquenessof solutions to this problem with estimates for the parabolic maximal functionof the spatial gradient of the solution.
AB - We consider a class of initial-boundary value problems for the heatequation on (0, T) x Ω with Ω a bounded Lipschitz domain in Rn. On thelateral boundary, (0, T) x Ω = ∑T, we specify (α, ∇v) where ∇v denotesthe spatial gradient of the solution and α: ∑T→(x: ∣x∣u = 1) is a continuousvector field satisfying (α, v) ≥ p > 0 with v the unit normal to Ω. On theinitial surface, (0) xΩ, we require that the solution vanish. The lateral datais taken from Lp∑T(I.t) ¦ For p ϵ (2 - ϵ, ∞), we show existence and uniquenessof solutions to this problem with estimates for the parabolic maximal functionof the spatial gradient of the solution.
KW - Heat equation
KW - Initial-boundary value problems
KW - Nonsmooth domains
UR - http://www.scopus.com/inward/record.url?scp=84966200319&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84966200319&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-1989-0987608-5
DO - 10.1090/S0002-9939-1989-0987608-5
M3 - Article
AN - SCOPUS:84966200319
SN - 0002-9939
VL - 107
SP - 237
EP - 250
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 1
ER -