Oblique derivative problems for elliptic and parabolic equations

Gary Lieberman

doi:10.3934/cpaa.2013.12.2409

Article Contents

2013, Volume 12, Issue 6: 2409-2444. Doi: 10.3934/cpaa.2013.12.2409

This issue Previous Article Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition Next Article A Brezis-Nirenberg result for non-local critical equations in low dimension

Oblique derivative problems for elliptic and parabolic equations

Gary Lieberman^1,

1.
Department of Mathematics, Iowa State University, Ames, IA 50011

Received: March 31, 2012

Revised: September 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

These notes are based on a series of lectures given by the author at the summer school of Partial Differential Equations at East China Normal University, Shanghai, July 18 through August 3, 2011. In these notes, we present information about linear oblique derivative problems for parabolic equations and nonlinear oblique derivative problems for elliptic equations. For the most part, all the theorems are true for both parabolic and elliptic problems provided we make some simple changes in the statements of the theorems to take into account the differences between the two types of equations, but we won't try to provide complete statements of results for the two classes of equations. Instead, we focus on presenting the basic techniques for these problems. Moreover, we only study second order equations, so that the maximum principle can be applied.

Keywords:

Mathematics Subject Classification: Primary: 35K20 , 35J25; Secondary: 35B45, 35J65.

Citation:

Related Papers

Cited by

References

[1]	L. A. Caffarelli, Interior estimates for solutions of fully nonlinear equations, Ann. Math., 130 (1989), 189-213.doi: 10.2307/1971480.
[2]	L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.doi: 10.2307/2152752.
[3]	L. A. Caffarelli, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math., 45 (1992), 1141-1151.doi: 10.1002/cpa.3160450905.
[4]	Ph. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampére operator, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 443-457.
[5]	Ph. Delanoë, Erratum to: "Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampére operator'', Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 849-850.doi: 10.1016/j.anihpc.2007.03.001.
[6]	G. C. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations, J. Partial Differential Equations, 1 (1988), 12-42.
[7]	D. Gilbarg and L. Hörmander, Intermediate Schauder estimates, Arch. Rational Mech. Anal., 74 (1980), 297-318.doi: 10.1007/BF00249677.
[8]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Classics in Mathematics (Reprint of the 1998 edition), Springer-Verlag, Berlin, 2001.
[9]	W. A. Kirk and J. Caristi, Mapping theorems in metric and Banach spaces, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys., 23 (1979), 891-894.
[10]	N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, [Russian], Sibirsk. Mat. Z., 17 (1976), 290-303, 478. English transl. in Sib. Math. J., 17 (1976), 226-236.
[11]	N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, [Russian], Izv. Akad. Nauk SSSR, 46 (1982), 487-523. English transl. in Math. USSR-Izv., 20 (1983), (459-493).
[12]	N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, [Russian], Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161-175, 239. English transl. in Math. USSR-Izv., 16 (1981), 155-164.
[13]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural$'$tseva, "Linear and Quasilinear Equations of Parabolic Type,'' [Russian], Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967.
[14]	O. A. Ladyzhenskaya and N. N. Ural$'$tseva, "Linear and Quasilinear Elliptic Equations,'' Izdat. "Nauka'', Moscow, 1964 [Russian]. English translation: Academic Press, New York, 1968.
[15]	G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions, Trans. Amer. Math. Soc., 273 (1982), 753-765.doi: 10.2307/1999940.
[16]	G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions II, J. Funct. Anal., 56 (1984), 210-219.doi: 10.1016/0022-1236(84)90087-9.
[17]	G. M. Lieberman, The Perron process applied to oblique derivative problems, Adv. Math., 55 (1985), 151-172.doi: 10.1016/0001-8708(85)90019-2.
[18]	G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Anal. Appl., 113 (1986), 422-440.doi: 10.1016/0022-247x(86)90314-8.
[19]	G. M. Lieberman, Intermediate Schauder estimates for oblique derivative problems, Arch. Rat. Mech. Anal., 93 (1986), 129-134.doi: 10.1007/BF00279956.
[20]	G. M. Lieberman, Oblique derivative problems in Lipschitz domains. I. Continuous boundary data, Boll. Unione. Mat. Ital., (7) (1987), 1185-1210.
[21]	G. M. Lieberman, The conormal derivative problem for parabolic equations, Indiana Univ. Math. J., 37 (1988), 23-72.doi: 10.1512/iumj.1988.37.37002.
[22]	G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. III. the tusk condition, Appl. Anal., 33 (1989), 25-43.doi: 10.1080/00036818908839859.
[23]	G. M. Lieberman, On the Hölder gradient estimate for solutions of nonlinear elliptic and parabolic oblique derivative problems, Comm. Partial Differential Equations, 15 (1990), 515-523.doi: 10.1080/03605309908820696.
[24]	G. M. Lieberman, "Second Order Parabolic Differential Equations,'' World Scientific, River Edge, NJ, 1996.
[25]	G. M. Lieberman, The maximum principle for equations with composite coefficients, Electronic J. Differential Equations, 2000 (2000), No. 38, 17 pp.
[26]	G. M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains, J. Differential Equations, 173 (2001), 178-211.doi: 10.1006/jdeq.2000.3939.
[27]	G. M. Lieberman, Higher regularity for nonlinear oblique derivative problems in Lipschitz domains, Ann. Scuola Norm. Sup. Pisa, 1 (2002), 111-151.
[28]	G. M. Lieberman, A new, simple existence theorem for fully nonlinear elliptic equations, Comm. Appl. Nonlinear Anal., 19 (2012), 1-13.
[29]	G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc., 295 (1986), 509-546.doi: 10.2307/2000050.
[30]	X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.doi: 10.1007/s00205-005-0362-9.
[31]	N. S. Nadirashvili, A lemma on the inner derivative, and the uniqueness of the solution of the second boundary value problem for second order elliptic equations, [Russian] Dokl. Akad. Nauk SSSR, 261 (1981), 804-808. English transl. in Soviet Math. Dokl., 24, (1981), 598-601.
[32]	N. S. Nadirashvili, On the question of the uniqueness of the solution of the second boundary value problem for second order elliptic equations, [Russian], Mat. Sb., 122 (1983), 341-359. English transl. in Math. USSR-Sb., 50 (1985), 325-341.
[33]	N. S. Nadirashvili, On a problem with an oblique derivative, [Russian], Mat. Sb. (N.S.), 127 (1985), 398-416. English transl. in Math. USSR-Sb., 55 (1986), 397-414.
[34]	N. S. Nadirashvili, Some estimates in a problem with an oblique derivative, [Russian], Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 1082-1090. Enlgish transl. in Math. USSR-Izv., 33 (1989), 403-411.
[35]	G. T. von Nessi, On the second boundary value problem for a class of modified-Hessian equations, Comm. Partial Differential Equations, 35 (2010), 745-785.doi: 10.1080/03605301003632317.
[36]	J. Pipher, Oblique derivative problems for the Laplacian in Lipschitz domains, Rev. Mat. Iberoamericana, 3 (1987), 455-472.doi: 10.4171/RMI/59.
[37]	A. V. Pogorelov, "Monge-Ampère Equations of Elliptic Type,'' Translated from the first Russian edition by Leo F. Boron with the assistance of Albert L. Rabenstein and Richard C. Bollinger, P. Noordhoff Ltd., Groningen, 1964.
[38]	M. V. Safonov, Classical solutions of nonlinear elliptic equations of second order, Izv. Akad. Nauk USSR, 52 (1988), 1272-1287. English transl. in Math. USSR-Izv., 33 (1989), 597-612.
[39]	N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.
[40]	N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math., 61 (1980), 67-79.doi: 10.1007/BF01389895.
[41]	N. S. Trudinger, Boundary value problems for fully nonlinear elliptic equations, in "Miniconference on Nonlinear Analysis,'' (Proc. Centre Math. Anal. Austral. Nat. Univ., 8 (1984), 65-83.
[42]	N. S. Trudinger, "Lectures on Nonlinear Second Order Elliptic Equations,'' Nankai Institute of Mathematics Lecture Notes, 1985.
[43]	N. S. Trudinger and X.-J. Wang, On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 143-174.
[44]	K. Tso, On an Aleksandrov-Bakel$'$man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations, 10 (1985), 543-553.doi: 10.1080/03605308508820388.
[45]	N. N. Uraltseva, Gradient estimates for solutions of nonlinear parabolic oblique boundary problem, in Contemp. Math., 127, Amer. Math. Soc., Providence, RI, (1992), 119-130.doi: 10.1090/conm/127/1155414.
[46]	J. Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 507-575.
[47]	J. Urbas, Nonlinear oblique boundary value problems for two-dimensional curvature equations, Adv. Differential Equations, 1 (1996), 301-336.
[48]	J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math., 487 (1997), 115-124.doi: 10.1515/crll.1997.487.115.
[49]	J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type, Calc. Var. Partial Differential Equations, 7 (1998), 19-39.doi: 10.1007/s005260050097.
[50]	J. Urbas, The second boundary value problem for a class of Hessian equations, Comm. Partial Differential Equations, 26 (2001), 859-882.doi: 10.1081/PDE-100002381.
[51]	J. G. Wolfson, Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation, J. Differential Geom., 46 (1997), 335-373.