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Optimal regularity of lower dimensional obstacle problems. (Russian, English) Zbl 1108.35038
J. Math. Sci., New York 132, No. 3, 274-284 (2006); translation from Zap. Nauchn. Semin. POMI 310, 49-66, 226 (2004).
The authors investigate the so-called “boundary obstacle problem”. They prove that solutions to this 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 tools within 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.

35J20 Variational methods for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35R35 Free boundary problems for PDEs
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