Regularity properties of a free boundary near contact points with the fixed boundary

Henrik Shahgholian, Nina Uraltseva

Результат исследований: Научные публикации в периодических изданияхстатья

23 Цитирования (Scopus)

Аннотация

In the upper half of the unit ball B+ = {|x| < 1, x1 > 0}, let u and Ω (a domain in R+ n = {x ∈ Rn: x1 > 0}) solve the following overdetermined problem: Δu = χΩ in B+, u = |∇u| = 0 in B+ / Ω, u = 0 on ∏ ∩ B, where B is the unit ball with center at the origin, χΩ denotes the characteristic function of Ω, ∏ = {X1 = 0}, n ≥ 2, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if u(0) = |∇u(0)| = 0, then ∂Ω∩ Br0 is the graph of a C1-function over ∏ ∩ Br0. The C1-norm depends on the dimension and sup B+ |u|. The result is extended to general subdomains of the unit ball with C3-boundary.

Язык оригиналаанглийский
Страницы (с-по)1-34
Число страниц34
ЖурналDuke Mathematical Journal
Том116
Номер выпуска1
DOI
СостояниеОпубликовано - 15 янв 2003

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  • Математика (все)

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