Geodesic
On properties of the free boundary in the neighborhood of contact with the given boundary
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 303-312

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For the simpliest elliptic obstacle problem, the behavior of the free boundary in the vicinity of the points where it contacts the prescribed boundary of the domain is studied. The eairlier result concerning the $C^1$ regularity of the boundary $\partial\mathscr N$ of the noncoincidence set is strengthened. It is proved that the assumed earlier Lipschitz condition on $\partial\mathscr N$ can be omitted. 
@article{ZNSL_1997_249_a14,
     author = {N. N. Ural'tseva},
     title = {On properties of the free boundary in the neighborhood of contact with the given boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {303--312},
     publisher = {mathdoc},
     volume = {249},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_249_a14/}
}
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N. N. Ural'tseva. On properties of the free boundary in the neighborhood of contact with the given boundary. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 303-312. http://geodesic.mathdoc.fr/item/ZNSL_1997_249_a14/