Higher regularity for nonlinear oblique derivative problems in Lipschitz domains

Gary M. Lieberman

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 111-151
  • ISSN: 0391-173X

Abstract

top
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.

How to cite

top

Lieberman, Gary M.. "Higher regularity for nonlinear oblique derivative problems in Lipschitz domains." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 111-151. <http://eudml.org/doc/84459>.

@article{Lieberman2002,
abstract = {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.},
author = {Lieberman, Gary M.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {111-151},
publisher = {Scuola normale superiore},
title = {Higher regularity for nonlinear oblique derivative problems in Lipschitz domains},
url = {http://eudml.org/doc/84459},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Lieberman, Gary M.
TI - Higher regularity for nonlinear oblique derivative problems in Lipschitz domains
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 111
EP - 151
AB - 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.
LA - eng
UR - http://eudml.org/doc/84459
ER -

References

top
  1. [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. Zbl0860.35046MR1372547
  2. [2] E. A. Baderko, Schauder estimates for oblique derivative problems, C. R. Acad. Sci. Paris Sér. I. Math. 326 (1998), 1377-1380. Zbl0912.35033MR1649177
  3. [3] L. A. Caffarelli, Interior estimates for fully nonlinear equations, Ann. Math. 130 (1989), 189-213. Zbl0692.35017MR1005611
  4. [4] G. C. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations 1 (1988), 12-42. Zbl0699.35152MR985445
  5. [5] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 25 (1982), 333-363. Zbl0469.35022MR649348
  6. [6] D. Gilbarg – L. Hörmander, Intermediate Schauder theory, Arch. Rational Mech. Anal. 74 (1980), 297-318. Zbl0454.35022MR588031
  7. [7] D. Gilbarg – N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin-New York-Heidelberg, 1977. Second Ed., 1983. Zbl0361.35003MR737190
  8. [8] G. Giraud, Nouvelle méthode pour traiter certains problèmes relatifs aux èquations du type elliptique, J. Math. Pures Appl. 18 (1939), 111-143. Zbl65.1277.02MR335JFM65.1277.02
  9. [9] J. Kovats, Fully nonlinear elliptic equations and the Dini condition, Comm. Partial Differential Equations 22 (1997), 1911-1927. Zbl0899.35036MR1629506
  10. [10] J. Kovats, Dini-Campanato spaces and applications to nonlinear elliptic equations, Electron. J. Differential Equations 1999 (1999), no. 37, 1-20. Zbl0926.35025MR1713596
  11. [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. Zbl0578.35024MR688919
  12. [12] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions, Trans. Amer. Math. Soc. 273 (1982), 753-765. Zbl0498.35039MR667172
  13. [13] G. M. Lieberman, The Perron process applied to oblique derivative problems, Adv. Math. 55 (1985), 161-172. Zbl0567.35027MR772613
  14. [14] G. M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), 329-352. Zbl0535.35028MR779924
  15. [15] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic equations of second order, J. Math. Anal. Appl. 113 (1986), 422-440. Zbl0609.35021MR826642
  16. [16] G. M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), 167-229. Zbl0589.35036MR818099
  17. [17] G. M. Lieberman, Oblique derivative problems in Lipschitz domains, Boll. Un. Mat. Ital. (7) 1-B (1987), 1185-1210. Zbl0637.35028MR923448
  18. [18] G. M. Lieberman, Optimal Hölder regularity for mixed boundary value problems, J. Math. Anal. Appl. 143 (1989), 572-586. Zbl0698.35034MR1022556
  19. [19] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations III. The tusk conditions, Appl. Anal. 33 (1989), 25-43. Zbl0702.35109MR1013451
  20. [20] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific, Singapore, 1996. Zbl0884.35001MR1465184
  21. [21] G. M. Lieberman, The maximum principle for equations with composite coefficients, Electron. J. Differential Equations, 2000 (2000) no. 38, 1-17. Zbl0952.35025MR1764710
  22. [22] G. M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains, J. Differential Equations 173 (2001), 178-211. Zbl0994.35042MR1836250
  23. [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. Zbl0619.35047MR833695
  24. [24] M. I. Matiichuk – S. D. Éidel ' 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. Zbl0296.35043MR348271
  25. [25] J. H. Michael, Barriers for uniformly elliptic equations and the exterior cone condition, J. Math. Anal. Appl. 79 (1981), 203-217. Zbl0454.35002MR603385
  26. [26] K. Miller, Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications, Ann. Mat. Pura Appl. (4) 90 (1971), 297-329. Zbl0231.35004MR316884
  27. [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. Zbl0563.35026MR721393
  28. [28] J. Pipher, Oblique derivative problems for the Laplacian in Lipschitz domains, Rev. Mat. Iberoamericana 3 (1987), 455-472. Zbl0686.35028MR996827
  29. [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. Zbl0682.35048MR984219
  30. [30] M. V. Safonov, On the oblique derivative problem for second order elliptic equations, Comm. Partial Differential Equations 20 (1995), 1349-1367. Zbl0841.35035MR1335754
  31. [31] E. Sperner, Jr., Schauder’s existence theorem for α -Dini continuous data, Ark. Mat. 19 (1981), 193-216. Zbl0506.35028MR650493
  32. [32] N. S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc. 278 (1983), 751-769. Zbl0518.35036MR701522
  33. [33] N. S. Trudinger, Hölder gradient estimates for fully nonlinear ellipotic equations, Proc. Roy. Soc. Edinburgh 108A (1988), 57-65. Zbl0653.35026MR931007
  34. [34] N. N. Ural ' 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. Zbl0770.35034MR1155414

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.