Geodesic
Optimal regularity of lower dimensional obstacle problems
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 49-66

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In this paper we prove that solutions to the “boundary obstacle problem” have the optimal regularity, $C^{1,1/2}$, in any space dimension. This bound depends only on the local $L^2$ norm of the solution. Main ingredients in the proof are the quasiconvexity of the solution and a monotonicity formula for an appropriate weighted average of the local energy of the normal derivative of the solution. 
@article{ZNSL_2004_310_a2,
     author = {I. Athanasopoulos and L. A. Caffarelli},
     title = {Optimal regularity of lower dimensional obstacle problems},
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     volume = {310},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_310_a2/}
}
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I. Athanasopoulos; L. A. Caffarelli. Optimal regularity of lower dimensional obstacle problems. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 49-66. http://geodesic.mathdoc.fr/item/ZNSL_2004_310_a2/