# American Institute of Mathematical Sciences

November  2013, 12(6): 2409-2444. doi: 10.3934/cpaa.2013.12.2409

## Oblique derivative problems for elliptic and parabolic equations

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

Received  April 2012 Revised  October 2012 Published  May 2013

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.
Citation: Gary Lieberman. Oblique derivative problems for elliptic and parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2409-2444. doi: 10.3934/cpaa.2013.12.2409
##### References:
 [1] L. A. Caffarelli, Interior estimates for solutions of fully nonlinear equations,, Ann. Math., 130 (1989), 189.  doi: 10.2307/1971480.  Google Scholar [2] L. A. Caffarelli, The regularity of mappings with a convex potential,, {J. Amer. Math. Soc.}, 5 (1992), 99.  doi: 10.2307/2152752.  Google Scholar [3] L. A. Caffarelli, Boundary regularity of maps with convex potentials,, {Comm. Pure Appl. Math.}, 45 (1992), 1141.  doi: 10.1002/cpa.3160450905.  Google Scholar [4] Ph. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampére operator,, {Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 8 (1991), 443.   Google Scholar [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. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 849.  doi: 10.1016/j.anihpc.2007.03.001.  Google Scholar [6] G. C. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations,, {J. Partial Differential Equations}, (1988), 12.   Google Scholar [7] D. Gilbarg and L. Hörmander, Intermediate Schauder estimates,, {Arch. Rational Mech. Anal.}, (1980), 297.  doi: 10.1007/BF00249677.  Google Scholar [8] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Classics in Mathematics (Reprint of the 1998 edition), (1998).   Google Scholar [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.   Google Scholar [10] N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, [Russian],, {Sibirsk. Mat. \v Z.}, 17 (1976), 290.   Google Scholar [11] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, [Russian],, {Izv. Akad. Nauk SSSR}, 46 (1982), 487.   Google Scholar [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.   Google Scholar [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, (1967).   Google Scholar [14] O. A. Ladyzhenskaya and N. N. Ural$'$tseva, "Linear and Quasilinear Elliptic Equations,'', Izdat., (1964).   Google Scholar [15] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions,, Trans. Amer. Math. Soc., 273 (1982), 753.  doi: 10.2307/1999940.  Google Scholar [16] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions II,, {J. Funct. Anal.}, 56 (1984), 210.  doi: 10.1016/0022-1236(84)90087-9.  Google Scholar [17] G. M. Lieberman, The Perron process applied to oblique derivative problems,, {Adv. Math.}, 55 (1985), 151.  doi: 10.1016/0001-8708(85)90019-2.  Google Scholar [18] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order,, {J. Math. Anal. Appl.}, 113 (1986), 422.  doi: 10.1016/0022-247x(86)90314-8.  Google Scholar [19] G. M. Lieberman, Intermediate Schauder estimates for oblique derivative problems,, {Arch. Rat. Mech. Anal.}, 93 (1986), 129.  doi: 10.1007/BF00279956.  Google Scholar [20] G. M. Lieberman, Oblique derivative problems in Lipschitz domains. I. Continuous boundary data,, Boll. Unione. Mat. Ital., (7) (1987), 1185.   Google Scholar [21] G. M. Lieberman, The conormal derivative problem for parabolic equations,, {Indiana Univ. Math. J.}, 37 (1988), 23.  doi: 10.1512/iumj.1988.37.37002.  Google Scholar [22] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. III. the tusk condition,, {Appl. Anal.}, 33 (1989), 25.  doi: 10.1080/00036818908839859.  Google Scholar [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.  doi: 10.1080/03605309908820696.  Google Scholar [24] G. M. Lieberman, "Second Order Parabolic Differential Equations,'', World Scientific, (1996).   Google Scholar [25] G. M. Lieberman, The maximum principle for equations with composite coefficients,, {Electronic J. Differential Equations}, 2000 (2000).   Google Scholar [26] G. M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains,, {J. Differential Equations}, 173 (2001), 178.  doi: 10.1006/jdeq.2000.3939.  Google Scholar [27] G. M. Lieberman, Higher regularity for nonlinear oblique derivative problems in Lipschitz domains,, {Ann. Scuola Norm. Sup. Pisa}, 1 (2002), 111.   Google Scholar [28] G. M. Lieberman, A new, simple existence theorem for fully nonlinear elliptic equations,, {Comm. Appl. Nonlinear Anal.}, 19 (2012), 1.   Google Scholar [29] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations,, {Trans. Amer. Math. Soc.}, 295 (1986), 509.  doi: 10.2307/2000050.  Google Scholar [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.  doi: 10.1007/s00205-005-0362-9.  Google Scholar [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.   Google Scholar [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.   Google Scholar [33] N. S. Nadirashvili, On a problem with an oblique derivative, [Russian],, {Mat. Sb. (N.S.)}, 127 (1985), 398.   Google Scholar [34] N. S. Nadirashvili, Some estimates in a problem with an oblique derivative, [Russian],, {Izv. Akad. Nauk SSSR Ser. Mat.}, 52 (1988), 1082.   Google Scholar [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.  doi: 10.1080/03605301003632317.  Google Scholar [36] J. Pipher, Oblique derivative problems for the Laplacian in Lipschitz domains,, {Rev. Mat. Iberoamericana}, 3 (1987), 455.  doi: 10.4171/RMI/59.  Google Scholar [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, (1964).   Google Scholar [38] M. V. Safonov, Classical solutions of nonlinear elliptic equations of second order,, {Izv. Akad. Nauk USSR}, 52 (1988), 1272.   Google Scholar [39] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations,, {Comm. Pure Appl. Math.}, 20 (1967), 721.   Google Scholar [40] N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations,, {Invent. Math.}, 61 (1980), 67.  doi: 10.1007/BF01389895.  Google Scholar [41] N. S. Trudinger, Boundary value problems for fully nonlinear elliptic equations,, in, 8 (1984), 65.   Google Scholar [42] N. S. Trudinger, "Lectures on Nonlinear Second Order Elliptic Equations,'', Nankai Institute of Mathematics Lecture Notes, (1985).   Google Scholar [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.   Google Scholar [44] K. Tso, On an Aleksandrov-Bakel$'$man type maximum principle for second-order parabolic equations,, {Comm. Partial Differential Equations}, 10 (1985), 543.  doi: 10.1080/03605308508820388.  Google Scholar [45] N. N. Uraltseva, Gradient estimates for solutions of nonlinear parabolic oblique boundary problem,, in Contemp. Math., 127 (1992), 119.  doi: 10.1090/conm/127/1155414.  Google Scholar [46] J. Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 12 (1995), 507.   Google Scholar [47] J. Urbas, Nonlinear oblique boundary value problems for two-dimensional curvature equations,, {Adv. Differential Equations}, 1 (1996), 301.   Google Scholar [48] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, {J. Reine Angew. Math.}, 487 (1997), 115.  doi: 10.1515/crll.1997.487.115.  Google Scholar [49] J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type,, {Calc. Var. Partial Differential Equations}, 7 (1998), 19.  doi: 10.1007/s005260050097.  Google Scholar [50] J. Urbas, The second boundary value problem for a class of Hessian equations,, Comm. Partial Differential Equations, 26 (2001), 859.  doi: 10.1081/PDE-100002381.  Google Scholar [51] J. G. Wolfson, Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation,, {J. Differential Geom.}, 46 (1997), 335.   Google Scholar

show all references

##### References:
 [1] L. A. Caffarelli, Interior estimates for solutions of fully nonlinear equations,, Ann. Math., 130 (1989), 189.  doi: 10.2307/1971480.  Google Scholar [2] L. A. Caffarelli, The regularity of mappings with a convex potential,, {J. Amer. Math. Soc.}, 5 (1992), 99.  doi: 10.2307/2152752.  Google Scholar [3] L. A. Caffarelli, Boundary regularity of maps with convex potentials,, {Comm. Pure Appl. Math.}, 45 (1992), 1141.  doi: 10.1002/cpa.3160450905.  Google Scholar [4] Ph. Delanoë, Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampére operator,, {Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 8 (1991), 443.   Google Scholar [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. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 849.  doi: 10.1016/j.anihpc.2007.03.001.  Google Scholar [6] G. C. Dong, Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations,, {J. Partial Differential Equations}, (1988), 12.   Google Scholar [7] D. Gilbarg and L. Hörmander, Intermediate Schauder estimates,, {Arch. Rational Mech. Anal.}, (1980), 297.  doi: 10.1007/BF00249677.  Google Scholar [8] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'', Classics in Mathematics (Reprint of the 1998 edition), (1998).   Google Scholar [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.   Google Scholar [10] N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, [Russian],, {Sibirsk. Mat. \v Z.}, 17 (1976), 290.   Google Scholar [11] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, [Russian],, {Izv. Akad. Nauk SSSR}, 46 (1982), 487.   Google Scholar [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.   Google Scholar [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, (1967).   Google Scholar [14] O. A. Ladyzhenskaya and N. N. Ural$'$tseva, "Linear and Quasilinear Elliptic Equations,'', Izdat., (1964).   Google Scholar [15] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions,, Trans. Amer. Math. Soc., 273 (1982), 753.  doi: 10.2307/1999940.  Google Scholar [16] G. M. Lieberman, Solvability of quasilinear elliptic equations with nonlinear boundary conditions II,, {J. Funct. Anal.}, 56 (1984), 210.  doi: 10.1016/0022-1236(84)90087-9.  Google Scholar [17] G. M. Lieberman, The Perron process applied to oblique derivative problems,, {Adv. Math.}, 55 (1985), 151.  doi: 10.1016/0001-8708(85)90019-2.  Google Scholar [18] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order,, {J. Math. Anal. Appl.}, 113 (1986), 422.  doi: 10.1016/0022-247x(86)90314-8.  Google Scholar [19] G. M. Lieberman, Intermediate Schauder estimates for oblique derivative problems,, {Arch. Rat. Mech. Anal.}, 93 (1986), 129.  doi: 10.1007/BF00279956.  Google Scholar [20] G. M. Lieberman, Oblique derivative problems in Lipschitz domains. I. Continuous boundary data,, Boll. Unione. Mat. Ital., (7) (1987), 1185.   Google Scholar [21] G. M. Lieberman, The conormal derivative problem for parabolic equations,, {Indiana Univ. Math. J.}, 37 (1988), 23.  doi: 10.1512/iumj.1988.37.37002.  Google Scholar [22] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. III. the tusk condition,, {Appl. Anal.}, 33 (1989), 25.  doi: 10.1080/00036818908839859.  Google Scholar [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.  doi: 10.1080/03605309908820696.  Google Scholar [24] G. M. Lieberman, "Second Order Parabolic Differential Equations,'', World Scientific, (1996).   Google Scholar [25] G. M. Lieberman, The maximum principle for equations with composite coefficients,, {Electronic J. Differential Equations}, 2000 (2000).   Google Scholar [26] G. M. Lieberman, Pointwise estimates for oblique derivative problems in nonsmooth domains,, {J. Differential Equations}, 173 (2001), 178.  doi: 10.1006/jdeq.2000.3939.  Google Scholar [27] G. M. Lieberman, Higher regularity for nonlinear oblique derivative problems in Lipschitz domains,, {Ann. Scuola Norm. Sup. Pisa}, 1 (2002), 111.   Google Scholar [28] G. M. Lieberman, A new, simple existence theorem for fully nonlinear elliptic equations,, {Comm. Appl. Nonlinear Anal.}, 19 (2012), 1.   Google Scholar [29] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations,, {Trans. Amer. Math. Soc.}, 295 (1986), 509.  doi: 10.2307/2000050.  Google Scholar [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.  doi: 10.1007/s00205-005-0362-9.  Google Scholar [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.   Google Scholar [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.   Google Scholar [33] N. S. Nadirashvili, On a problem with an oblique derivative, [Russian],, {Mat. Sb. (N.S.)}, 127 (1985), 398.   Google Scholar [34] N. S. Nadirashvili, Some estimates in a problem with an oblique derivative, [Russian],, {Izv. Akad. Nauk SSSR Ser. Mat.}, 52 (1988), 1082.   Google Scholar [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.  doi: 10.1080/03605301003632317.  Google Scholar [36] J. Pipher, Oblique derivative problems for the Laplacian in Lipschitz domains,, {Rev. Mat. Iberoamericana}, 3 (1987), 455.  doi: 10.4171/RMI/59.  Google Scholar [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, (1964).   Google Scholar [38] M. V. Safonov, Classical solutions of nonlinear elliptic equations of second order,, {Izv. Akad. Nauk USSR}, 52 (1988), 1272.   Google Scholar [39] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations,, {Comm. Pure Appl. Math.}, 20 (1967), 721.   Google Scholar [40] N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations,, {Invent. Math.}, 61 (1980), 67.  doi: 10.1007/BF01389895.  Google Scholar [41] N. S. Trudinger, Boundary value problems for fully nonlinear elliptic equations,, in, 8 (1984), 65.   Google Scholar [42] N. S. Trudinger, "Lectures on Nonlinear Second Order Elliptic Equations,'', Nankai Institute of Mathematics Lecture Notes, (1985).   Google Scholar [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.   Google Scholar [44] K. Tso, On an Aleksandrov-Bakel$'$man type maximum principle for second-order parabolic equations,, {Comm. Partial Differential Equations}, 10 (1985), 543.  doi: 10.1080/03605308508820388.  Google Scholar [45] N. N. Uraltseva, Gradient estimates for solutions of nonlinear parabolic oblique boundary problem,, in Contemp. Math., 127 (1992), 119.  doi: 10.1090/conm/127/1155414.  Google Scholar [46] J. Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 12 (1995), 507.   Google Scholar [47] J. Urbas, Nonlinear oblique boundary value problems for two-dimensional curvature equations,, {Adv. Differential Equations}, 1 (1996), 301.   Google Scholar [48] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type,, {J. Reine Angew. Math.}, 487 (1997), 115.  doi: 10.1515/crll.1997.487.115.  Google Scholar [49] J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type,, {Calc. Var. Partial Differential Equations}, 7 (1998), 19.  doi: 10.1007/s005260050097.  Google Scholar [50] J. Urbas, The second boundary value problem for a class of Hessian equations,, Comm. Partial Differential Equations, 26 (2001), 859.  doi: 10.1081/PDE-100002381.  Google Scholar [51] J. G. Wolfson, Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation,, {J. Differential Geom.}, 46 (1997), 335.   Google Scholar
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