Vol. 9, No. 2, 2016

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Obstacle problem with a degenerate force term

Karen Yeressian

Vol. 9 (2016), No. 2, 397–437

We study the regularity of the free boundary at its intersection with the line {x1 = 0} in the obstacle problem

u = |x1|χ{u>0} in D,

where D 2 is a bounded domain such that D {x1 = 0}.

We obtain a uniform C1,1 bound on cubic blowups; we find all homogeneous global solutions; we prove the uniqueness of the blowup limit; we prove the convergence of the free boundary to the free boundary of the blowup limit; at the points with lowest Weiss balanced energy, we prove the convergence of the normal of the free boundary to the normal of the free boundary of the blowup limit and that locally the free boundary is a graph; and, finally, for a particular case we prove that the free boundary is not C1,α regular near to a degenerate point for any 0 < α < 1.

free boundary, obstacle problem, degenerate, blowup, regularity
Mathematical Subject Classification 2010
Primary: 35R35
Secondary: 35J60
Received: 25 October 2014
Revised: 27 October 2015
Accepted: 6 January 2016
Published: 24 March 2016
Karen Yeressian
Institute of Mathematics
University of Zurich
Winterthurerstrasse 190
CH-8057 Zurich