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On Serrin’s overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg Wei, Juncheng, 1968-
Description
In 1971, Serrin proved that the only bounded domain for which the overdetermined problem ∆u + f (u) = 0, u > 0 in Ω u = 0 on ∂Ω ∂νu = C on ∂Ω admits a solution is the bounded ball. In 1997, Berestycki, Caffarelli and Nirenberg considered the unbounded domain case, and proposed the following conjecture: If Serrin’s problem admits a solution and Ωc is connected, then Ω is either a half space, a cylinder B × RN−k, or complement of a ball or cylinder. In this talk, I shall discuss positive and negative answers to this conjecture.In particular, when Ω is an epigraph Ω = {xN > φ(x′)}, we show that (1) BCN conjecture is always true when N = 2, (2) BCN conjecture is true when 3 ≤ N ≤ 8 if ∂u > 0 (3) BCN conjecture is false when N ≥ 9. A key observation ∂xN is the connection between this problem and a one-phase free boundary problem.
Item Metadata
Title |
On Serrin’s overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2015-09-01T09:00
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Description |
In 1971, Serrin proved that the only bounded domain for which the overdetermined problem
∆u + f (u) = 0, u > 0 in Ω
u = 0 on ∂Ω
∂νu = C on ∂Ω
admits a solution is the bounded ball. In 1997, Berestycki, Caffarelli
and Nirenberg considered the unbounded domain case, and proposed
the following conjecture: If Serrin’s problem admits a solution and Ωc
is connected, then Ω is either a half space, a cylinder B × RN−k, or
complement of a ball or cylinder. In this talk, I shall discuss positive
and negative answers to this conjecture.In particular, when Ω is an
epigraph Ω = {xN > φ(x′)}, we show that (1) BCN conjecture is
always true when N = 2, (2) BCN conjecture is true when 3 ≤ N ≤ 8
if ∂u > 0 (3) BCN conjecture is false when N ≥ 9. A key observation ∂xN
is the connection between this problem and a one-phase free boundary problem.
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Extent |
53 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of British Columbia
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Series | |
Date Available |
2016-03-11
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0300074
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International