Abstract
In this paper, we prove that solutions to the “boundary obstacle problem” have the optimal regularity, C1,1/2, in any space dimension. This bound depends only on the local L2-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. Bibliography: 8 titles.
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REFERENCES
I. Athanasopoulos, “Regularity of the solution of an evolution problem with inequalities on the boundary,” Comm. P.D.E., 7, 1453–1465 (1982).
L. A. Caffarelli, “Further regularity for the Signorini problem,” Comm. P.D.E., 4, 1067–1075 (1979).
L. A. Caffarelli, “The obstacle problem revisited,” J. Fourier Anal. Appl., 4, 383–402 (1998).
G. Duvaut and J. L. Lions, Les Inequations en Mechanique et en Physique, Dunod, Paris (1972).
J. L. Lions and G. Stampacchia, “Variational inequalities,” Comm. Pure Appl. Math., 20, 493–519 (1967).
D. Richardson, Thesis, University of British Columbia (1978).
L. Silvestre, Thesis, University of Texas at Austin (in preparation).
N. N. Uraltseva, “On the regularity of solutions of variational inequalities,” Usp. Mat. Nauk, 42, 151–174 (1987).
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Dedicated to Nina Nikolaevna Uraltseva on the occasion of her 70th birthday
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 49–66.
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Athanasopoulos, I., Caffarelli, L.A. Optimal Regularity of Lower-Dimensional Obstacle Problems. J Math Sci 132, 274–284 (2006). https://doi.org/10.1007/s10958-005-0496-1
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DOI: https://doi.org/10.1007/s10958-005-0496-1