Duke Mathematical Journal

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

Henrik Shahgholian and Nina Uraltseva

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In the upper half of the unit ball $B\sp + =\{ |x|<1,x\sb 1>0\}$, let $u$ and $\Omega$ (a domain in $\mathbf {R}\sp n\sb + =\{x\in \mathbf {R}\sp n : x\sb 1>0\}$) solve the following overdetermined problem:

\Delta u =\chi_\Omega\quad \text{in}\ B^+, \qquad u=|\nabla u |=0 \quad \text{in}\ B^+\setminus \Omega, \qquad u=0 \quad \text{on}\ \Pi\cap B,

where $B$ is the unit ball with center at the origin, $\chi\sb \Omega$ denotes the characteristic function of $\Delta,\Pi=\{ x\sb 1=0\} ,n\geq 2$, 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 $u(0)=|\nabla u(0)|=0$, then $\partial\Delta\cap B\sb {r\sb 0}$ is the graph of a $C\sp 1$-function over $\Pi\cap B\sb (r\sb 0)$. The $C\sp 1$-norm depends on the dimension and $\sup\sb {B\sp +}|u|$. The result is extended to general subdomains of the unit ball with $C\sp 3$-boundary.

Article information

Duke Math. J. Volume 116, Number 1 (2003), 1-34.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 35R35: Free boundary problems
Secondary: 35B65: Smoothness and regularity of solutions 35J60: Nonlinear elliptic equations


Shahgholian, Henrik; Uraltseva, Nina. Regularity properties of a free boundary near contact points with the fixed boundary. Duke Math. J. 116 (2003), no. 1, 1--34. doi:10.1215/S0012-7094-03-11611-7. http://projecteuclid.org/euclid.dmj/1085598234.

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