TY - JOUR

T1 - Regularity properties of a free boundary near contact points with the fixed boundary

AU - Shahgholian, Henrik

AU - Uraltseva, Nina

PY - 2003/1/15

Y1 - 2003/1/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037440335&partnerID=8YFLogxK

U2 - 10.1215/S0012-7094-03-11611-7

DO - 10.1215/S0012-7094-03-11611-7

M3 - Article

AN - SCOPUS:0037440335

VL - 116

SP - 1

EP - 34

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 1

ER -