Degenerate variational inequalities with gradient constraints

Choe, Hi Jun; Shim, Yong Sun

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 22 (1995) no. 1, pp. 25-53

[Br1] H. Brezis - D. Kinderlehrer, The Smoothness of Solutions to Nonlinear Variational Inequalities. Indiana Univ. Math. J. 23 (1974), 831-844. | Zbl | MR

[Br2] H. Brezis - G. Stampacchia, Sur la regularite de la solution d'inequations elliptiques. Bull. Soc. Math. France 96 (1968), 153-180. | Zbl | MR | Numdam

[Caf] L. Caffarelli - N.M. Riviere, The Lipschitz Character of the Stress Tensor When Twisting an Elastic and Plastic Bar. Arch. Rational Mech. Anal. 69 (1979), 31-36. | Zbl | MR

[Can] P Cannarsa - H.M. Soner, On the Singularities of the Viscosity Solutions to Hamilton-Jacobi-Bellman Equations. Indiana Univ. Math. J. 36 (1987), 501-524. | Zbl | MR

[Ch1] H.J. Choe, A regularity theory for a more general class of quasilinear elliptic partial differential equations and obstacle problems. Arch. Rational Mech. Anal. 114 (1991), 383-394. | Zbl | MR

[Ch2] H.J. Choe - Y.S. Shim, On the variational inequalities for certain convex function classes. To appear in J. Differential Equations. | Zbl | MR

[Chi] M. Chipot, Variational Inequalities and Flow in Porous Media. Applied Mathematical Sciences, 52, Springer, Berlin 1984. | Zbl | MR

[DiB] E. Di Benedetto, C1,α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983), 827-850. | Zbl

[Eva] L.C. Evans, A Second Order Elliptic Equation with Gradient Constraint. Comm. Partial Differential Equations 4 (1979), 555-572. | Zbl | MR

[Fle] W.H. Fleming - P.E. Souganidis, Asymptotic series and the method of vanishing viscosity. Indiana Univ. Math. J. 35 (1986), 425-447. | Zbl | MR

[Ge1] C. Gerhardt, Regularity of Solutions of Nonlinear Variational Inequalities with a Gradient Bound as Constraint. Arch. Rational Mech. Anal. 58 (1975), 309-317. | Zbl | MR

[Ge2] C. Gerhardt, Regularity of Solutions of Nonlinear Variational Inequalities. Arch. Rational Mech. Anal. 52 (1973), 389-393. | Zbl | MR

[Gia] M. Giaquinta, Remarks on the regularity of weak solutions to some variational inequalities. Math. Z. 177 (1981), 15-31. | Zbl | MR | EuDML

[GT] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer-Verlag, Berlin 1983. | Zbl | MR

[Har] P. Hartman - G. Stampacchia, On some nonlinear elliptic differential functional equations. Acta Math. 115 (1971), 271-310. | Zbl | MR

[IK] H. Ishii - S. Koike, Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Comm. Partial Differential Equations 8 (1983), 317-346. | Zbl | MR

[Is1] H. Ishii, Perron's method for Hamilton-Jacobi equations. Preprint. | MR | Zbl

[Is2] H. Ishii, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of Eikonal type. Proc. Amer. Math. Soc. 100 (1987), 247-251. | Zbl | MR

[Jen] R. Jensen, Regularity for Elasto-Plastic Type Variational Inequality. Indiana Univ. Math. J. 32 (1983), 407-423. | Zbl | MR

[Kry] N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain. (Russian). Izv. Akad. Nauk SSSR 147 (1983), 75-108. | Zbl | MR

[Lad] O.A. Ladyzhenskaya - N.N. Ural'Tzeva, Linear and Quasilinear Elliptic Equations. Academic Press, 1968. | Zbl | MR

[Lew] J. Lewis, Regularity of derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. Math. J. 32 (1983), 849-858. | Zbl | MR

[Li1] G. Lieberman, Local and boundary regularity for some variational inequalities involving p-Laplacian-type operators. Preprint.

[Li2] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tzeva for elliptic equations, Comm. Partial Differential Equations 16 (1991), 311-362. | Zbl | MR

[Li3] G. Lieberman, Regularity of solutions to some degenerate double obstacle problems. Preprint. | MR | Zbl

[Lin] F.H. Lin - Y. Li, Boundary C1,α regularity for variational inequalities. Comm. Pure Appl. Math. XLIV (1991), 715-732. | Zbl | MR

[Lind] P. Lindquist, Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal. 12 (1988), 1245-1255. | MR

[Lio1] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Research Notes in Mathematics, vol. 69, Pitman Advanced Publishing Program, 1982. | Zbl | MR

[Lio2] P.L. Lions, Resolution de Problèmes Elliptiques Quasilinéaires. Arch. Rational Mech. Anal. 74 (1980), 335-353. | Zbl | MR

[Mic] J. Michael - W. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), 1427-1448. | Zbl | MR

[Mu1] J. Mu, Higher regularity of the solution to the p-Laplacian obstacle problem. Preprint. | MR | Zbl

[Mu2] J. Mu - W. Ziemer, Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations. Preprint. | MR | Zbl

[Nor] T. Norando, C1,α local regularity for a class of quasilinear elliptic variational inequalities. Boll. Un. Mat. Ital. 5 (1986), 281-291. | Zbl | MR

[To1] P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations. J. Differential Equations 51 (1984), 126-150. | Zbl | MR

[To2] P. Tolksdorff, Everywhere regularity for some quasilinear systems with a lack of ellipticity. Ann. Mat. Pura Appl. 134 (1983), 241-266. | Zbl | MR

[Uhl] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219-240. | Zbl | MR

[Wi] M. Wiegner, The C1,1 -character of solutions of second order elliptic equations with gradient constraints. Comm. Partial Differential Equations 6 (1981), 361-371. | Zbl | MR