The oblique derivative problem for the heat equation in lipschitz cylinders

Russell M. Brown

Research output: Contribution to journalArticlepeer-review


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 LpT(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.

Original languageEnglish
Pages (from-to)237-250
Number of pages14
JournalProceedings of the American Mathematical Society
Issue number1
StatePublished - Sep 1989


  • Heat equation
  • Initial-boundary value problems
  • Nonsmooth domains

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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