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Boundary regularity for linear and quasilinear variational inequalities

Published online by Cambridge University Press:  14 November 2011

Gary M. Lieberman
Gary M. Lieberman
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A.
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Synopsis

A method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 3-4 , 1989 , pp. 319 - 326
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Arkhipova, A. A.. Limiting smoothness of the solution of a nonstationary problem with one or two obstacles. Probl. Mat. Anal. 9 (1984), 149–157 (in Russian). English translation in J. Soviet Math. 35 (1986), 2257–2262.Google Scholar
2Brezis, H. and Kinderlehrer, D.. The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23 (1974), 837–844.CrossRefGoogle Scholar
3Caffarelli, L. A. and Kinderlehrer, D.. Potential methods in variational inequalities. J. Analyse Math. 37 (1980), 285–295.CrossRefGoogle Scholar
4Friedman, A.. Variational Principles and Free-Boundary Problems (New York: Wiley-Interscience, 1982).Google Scholar
5Gerhardt, C.. Global C1,1-regularity for solutions to quasilinear variationalinequalities. Arch. Rational Mech. Anal. 89 (1985), 83–92.CrossRefGoogle Scholar
6Giaquinta, M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud. 105 (Princeton: Princeton University Press, 1983).Google Scholar
7Gilbarg, D. and Hörmander, L.. Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74 (1980), 297–318.CrossRefGoogle Scholar
8Huisken, B.. Second-order boundary regularity for quasilinear variational inequalities. Manuscripta Math. 63 (1989), 333–342.CrossRefGoogle Scholar
9Huisken, G.. Capillarity surfaces over obstacles. Pacific J. Math. 117 (1985), 121–141.CrossRefGoogle Scholar
10Jensen, R.. Boundary regularity for variational inequalities. Indiana Univ. Math. J. 29 (1980), 495–504.CrossRefGoogle Scholar
11Lieberman, G. M.. Intermediate Schauder theory for second order parabolic equations. I.Estimates. J. Differential Equations 63 (1986), 1–31.CrossRefGoogle Scholar
12Lieberman, G. M.. Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations. Trans. Amer. Math. Soc. 304 (1987), 343–353.CrossRefGoogle Scholar
13Lieberman, G. M.. Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions. Ann. Mat. Pura Appl. (4) 138 (1987), 77–99, 397–398.CrossRefGoogle Scholar
14Lions, J. L.. Sur les problèmes aux limites du type derivée oblique. Ann. of Math. (2) 64 (1956), 207–239.CrossRefGoogle Scholar
15Rodrigues, J.-F.. Obstacle Problems in Mathematical Physics, North Holland Math.Stud. 134 (Amsterdam: North-Holland, 1987).CrossRefGoogle Scholar