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March  2018, 17(2): 413-428. doi: 10.3934/cpaa.2018023

## The regularity of some vector-valued variational inequalities with gradient constraints

 Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA

Received  April 2015 Revised  October 2015 Published  March 2018

We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar case and in the vector-valued case.

Citation: Mohammad Safdari. The regularity of some vector-valued variational inequalities with gradient constraints. Communications on Pure & Applied Analysis, 2018, 17 (2) : 413-428. doi: 10.3934/cpaa.2018023
##### References:
 [1] H. Brezis and M. Sibony, Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal., 41 (1971), 254-265. doi: 10.1007/BF00250529. Google Scholar [2] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France, 96 (1968), 153-180. Google Scholar [3] L. A. Caffarelli and N. M. Riviére, The Lipschitz character of the stress tensor, when twisting an elastic plastic bar, Arch. Rational Mech. Anal., 69 (1979), 31-36. doi: 10.1007/BF00248408. Google Scholar [4] L. C. Evans, A second-order elliptic equation with gradient constraint, Comm. Partial Differential Equations, 4 (1979), 555-572. doi: 10.1080/03605307908820103. Google Scholar [5] A. Friedman, Variational Principles And Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc. , New York, 1982, A Wiley-Interscience Publication. Google Scholar [6] C. Gerhardt, Regularity of solutions of nonlinear variational inequalities with a gradient bound as constraint, Arch. Rational Mech. Anal., 58 (1975), 309-315. doi: 10.1007/BF00250293. Google Scholar [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Google Scholar [8] H. Ishii and S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint, Comm. Partial Differential Equations, 8 (1983), 317-346. doi: 10.1080/03605308308820271. Google Scholar [9] R. Jensen, Regularity for elastoplastic type variational inequalities, Indiana Univ. Math. J., 32 (1983), 407-423. doi: 10.1512/iumj.1983.32.32030. Google Scholar [10] C. Mariconda and G. Treu, Gradient maximum principle for minima, J. Optim. Theory Appl., 112 (2002), 167-186. doi: 10.1023/A:1013052830852. Google Scholar [11] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N. J. , 1970. Google Scholar [12] T. N. Rozhkovskaya, Unilateral problems for elliptic systems with gradient constraints, in Partial differential equations, Part 1, 2 (Warsaw, 1990), vol. 2 of Banach Center Publ., 27, Part 1, Polish Acad. Sci., Warsaw, (1992), 425-445. Google Scholar [13] G. Treu and M. Vornicescu, On the equivalence of two variational problems, Calc. Var. Partial Differential Equations, 11 (2000), 307-319. doi: 10.1007/s005260000040. Google Scholar [14] M. Wiegner, The $C^{1, 1}$-character of solutions of second order elliptic equations with gradient constraint, Comm. Partial Differential Equations, 6 (1981), 361-371. doi: 10.1080/03605308108820181. Google Scholar

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##### References:
 [1] H. Brezis and M. Sibony, Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal., 41 (1971), 254-265. doi: 10.1007/BF00250529. Google Scholar [2] H. Brezis and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France, 96 (1968), 153-180. Google Scholar [3] L. A. Caffarelli and N. M. Riviére, The Lipschitz character of the stress tensor, when twisting an elastic plastic bar, Arch. Rational Mech. Anal., 69 (1979), 31-36. doi: 10.1007/BF00248408. Google Scholar [4] L. C. Evans, A second-order elliptic equation with gradient constraint, Comm. Partial Differential Equations, 4 (1979), 555-572. doi: 10.1080/03605307908820103. Google Scholar [5] A. Friedman, Variational Principles And Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc. , New York, 1982, A Wiley-Interscience Publication. Google Scholar [6] C. Gerhardt, Regularity of solutions of nonlinear variational inequalities with a gradient bound as constraint, Arch. Rational Mech. Anal., 58 (1975), 309-315. doi: 10.1007/BF00250293. Google Scholar [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Google Scholar [8] H. Ishii and S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint, Comm. Partial Differential Equations, 8 (1983), 317-346. doi: 10.1080/03605308308820271. Google Scholar [9] R. Jensen, Regularity for elastoplastic type variational inequalities, Indiana Univ. Math. J., 32 (1983), 407-423. doi: 10.1512/iumj.1983.32.32030. Google Scholar [10] C. Mariconda and G. Treu, Gradient maximum principle for minima, J. Optim. Theory Appl., 112 (2002), 167-186. doi: 10.1023/A:1013052830852. Google Scholar [11] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N. J. , 1970. Google Scholar [12] T. N. Rozhkovskaya, Unilateral problems for elliptic systems with gradient constraints, in Partial differential equations, Part 1, 2 (Warsaw, 1990), vol. 2 of Banach Center Publ., 27, Part 1, Polish Acad. Sci., Warsaw, (1992), 425-445. Google Scholar [13] G. Treu and M. Vornicescu, On the equivalence of two variational problems, Calc. Var. Partial Differential Equations, 11 (2000), 307-319. doi: 10.1007/s005260000040. Google Scholar [14] M. Wiegner, The $C^{1, 1}$-character of solutions of second order elliptic equations with gradient constraint, Comm. Partial Differential Equations, 6 (1981), 361-371. doi: 10.1080/03605308108820181. Google Scholar
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