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MiS Preprint
43/2006
The Free Boundary Near the Fixed Boundary for the Heat Equation
John Andersson
Abstract
We study the regularity of the free boundary, near contact points with the fixed boundary, for a parabolic free boundary problem: $$ \begin{array}{ll} \Delta u-\frac{\partial u}{\partial t}=\chi_{\{u\ne 0\}} & \textrm{in } B_r^+. \end{array} $$ We will show that the free boundary is a $C^1$ manifold up to the fixed boundary under certain regularity assumptions on the boundary data, the $C^1$ norm is uniform for a certain, and specified, subclass of solutions.
