Higher regularity for nonlinear oblique derivative problems in Lipschitz domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 1, p. 111-151

There is a long history of studying nonlinear boundary value problems for elliptic differential equations in a domain with sufficiently smooth boundary. In this paper, we show that the gradient of the solution of such a problem is continuous when a directional derivative is prescribed on the boundary of a Lipschitz domain for a large class of nonlinear equations under weak conditions on the data of the problem. The class of equations includes linear equations with fairly rough coefficients as well as Bellman equations.

@article{ASNSP_2002_5_1_1_111_0,
author = {Lieberman, Gary M.},
title = {Higher regularity for nonlinear oblique derivative problems in Lipschitz domains},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {1},
year = {2002},
pages = {111-151},
zbl = {1170.35423},
zbl = {pre02216749},
mrnumber = {1994804},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2002_5_1_1_111_0}
}

Lieberman, Gary M. Higher regularity for nonlinear oblique derivative problems in Lipschitz domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 1, pp. 111-151. http://www.numdam.org/item/ASNSP_2002_5_1_1_111_0/

[1] D. E. Apushkinskaya - A. I. Nazarov, Boundary estimates for the first-order derivatives of a solution to a nondivergent parabolic equation with composite right-hand side and coefficients of lower-order derivatives, Prob. Mat. Anal. 14 (1995), 3-27 [Russian]; English transl. in J. Math. Sci. 77 (1995), 3257-3276. | MR 1372547 | Zbl 0860.35046

[2] E. A. Baderko, Schauder estimates for oblique derivative problems, C. R. Acad. Sci. Paris Sér. I. Math. 326 (1998), 1377-1380. | MR 1649177 | Zbl 0912.35033

[3] L. A. Caffarelli, Interior estimates for fully nonlinear equations, Ann. Math. 130 (1989), 189-213. | MR 1005611 | Zbl 0692.35017

[4] G. C. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations 1 (1988), 12-42. | MR 985445 | Zbl 0699.35152

[5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 25 (1982), 333-363. | MR 649348 | Zbl 0469.35022

[6] D. Gilbarg - L. Hörmander, Intermediate Schauder theory, Arch. Rational Mech. Anal. 74 (1980), 297-318. | MR 588031 | Zbl 0454.35022

[7] D. Gilbarg - N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin-New York-Heidelberg, 1977. Second Ed., 1983. | MR 737190 | Zbl 0361.35003

[8] G. Giraud, Nouvelle méthode pour traiter certains problèmes relatifs aux èquations du type elliptique, J. Math. Pures Appl. 18 (1939), 111-143. | JFM 65.1277.02 | MR 335

[9] J. Kovats, Fully nonlinear elliptic equations and the Dini condition, Comm. Partial Differential Equations 22 (1997), 1911-1927. | MR 1629506 | Zbl 0899.35036

[10] J. Kovats, Dini-Campanato spaces and applications to nonlinear elliptic equations, Electron. J. Differential Equations 1999 (1999), no. 37, 1-20. | MR 1713596 | Zbl 0926.35025

[11] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR 47 (1983), 75-108 [Russian]; English transl. in Math.-USSR Izv. 22 (1984), 67-97. | MR 688919 | Zbl 0578.35024

[12] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions, Trans. Amer. Math. Soc. 273 (1982), 753-765. | MR 667172 | Zbl 0498.35039

[13] G. M. Lieberman, The Perron process applied to oblique derivative problems, Adv. Math. 55 (1985), 161-172. | MR 772613 | Zbl 0567.35027

[14] G. M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), 329-352. | MR 779924 | Zbl 0535.35028

[15] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic equations of second order, J. Math. Anal. Appl. 113 (1986), 422-440. | MR 826642 | Zbl 0609.35021

[16] G. M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), 167-229. | MR 818099 | Zbl 0589.35036

[17] G. M. Lieberman, Oblique derivative problems in Lipschitz domains, Boll. Un. Mat. Ital. (7) 1-B (1987), 1185-1210. | MR 923448 | Zbl 0637.35028

[18] G. M. Lieberman, Optimal Hölder regularity for mixed boundary value problems, J. Math. Anal. Appl. 143 (1989), 572-586. | MR 1022556 | Zbl 0698.35034

[19] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations III. The tusk conditions, Appl. Anal. 33 (1989), 25-43. | MR 1013451 | Zbl 0702.35109

[20] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, Singapore, 1996. | MR 1465184 | Zbl 0884.35001

[21] G. M. Lieberman, The maximum principle for equations with composite coefficients, Electron. J. Differential Equations, 2000 (2000) no. 38, 1-17. | MR 1764710 | Zbl 0952.35025

[22] G. M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains, J. Differential Equations 173 (2001), 178-211. | MR 1836250 | Zbl 0994.35042

[23] G. M. Lieberman - N. S. Trudinger, Nonlinear oblique boundary value problems for fully nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509-546. | MR 833695 | Zbl 0619.35047

[24] M. I. Matiichuk - S. D. Éidel${}^{\text{'}}$man, On the correctness of the problem of Dirichlet and Neumann for second-order parabolic equations with coefficients in Dini classes, Ukr. Mat. Zh. 26 (1974), 328-337 [Russian]; English transl. in Ukrainian Math. J. 26 (1974), 269-276. | MR 348271 | Zbl 0296.35043

[25] J. H. Michael, Barriers for uniformly elliptic equations and the exterior cone condition, J. Math. Anal. Appl. 79 (1981), 203-217. | MR 603385 | Zbl 0454.35002

[26] K. Miller, Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications, Ann. Mat. Pura Appl. (4) 90 (1971), 297-329. | MR 316884 | Zbl 0231.35004

[27] N. S. Nadirashvili, On the question of the uniqueness of the solution of the second boundary value problem for second order elliptic equations, Mat. Sb. (N. S.) 122 (164) (1983), 341-359 [Russian]; English transl. in Math. USSR-Sb. 50 (1985), 325-341. | MR 721393 | Zbl 0563.35026

[28] J. Pipher, Oblique derivative problems for the Laplacian in Lipschitz domains, Rev. Mat. Iberoamericana 3 (1987), 455-472. | MR 996827 | Zbl 0686.35028

[29] M. V. Safonov, On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk SSSR 52 (1988), 1272-1287. [Russian]; English transl. in Math. USSR-Izv. 33 (1989), 597-612. | MR 984219 | Zbl 0682.35048

[30] M. V. Safonov, On the oblique derivative problem for second order elliptic equations, Comm. Partial Differential Equations 20 (1995), 1349-1367. | MR 1335754 | Zbl 0841.35035

[31] E. Sperner, Jr., Schauder’s existence theorem for $\alpha$-Dini continuous data, Ark. Mat. 19 (1981), 193-216. | MR 650493 | Zbl 0506.35028

[32] N. S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc. 278 (1983), 751-769. | MR 701522 | Zbl 0518.35036

[33] N. S. Trudinger, Hölder gradient estimates for fully nonlinear ellipotic equations, Proc. Roy. Soc. Edinburgh 108A (1988), 57-65. | MR 931007 | Zbl 0653.35026

[34] N. N. Ural${}^{\text{'}}$tseva, Gradient estimates for solutions of nonlinear parabolic oblique boundary problem, in “Geometry and Nonlinear Partial Differential Equations”, American Mathematical Society, Providence, RI, 1992, pp. 119-130. | MR 1155414 | Zbl 0770.35034