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July  2016, 21(5): 1525-1566. doi: 10.3934/dcdsb.2016010

## Schauder estimates for singular parabolic and elliptic equations of Keldysh type

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

Received  October 2013 Revised  November 2015 Published  April 2016

We show that solutions of equations of the form $-u_t+D_{11}u+(x^1)D_{22}u = f$ (and also more general equations in any number of dimensions) satisfy simple Hölder estimates involving their derivatives. We also examine some pointwise properties for these solutions. Our results generalize those of Daskalopoulos and Lee, and Hong and Huang.
Citation: Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010
##### References:
 [1] A. D. Aleksandrov, Certain estimates for the Dirichlet problem, Dokl Akad. Nauk SSSR, 134 (1961), 1001-1004 [Russian]; English transl. in Soviet Math. Dokl. 1 (1961), 1151-1154.  Google Scholar [2] I. Ya. Bakel'man, Theory of quasilinear elliptic equations, Sibirsk. Mat. Zh., 2 (1961), 179-186 [Russian].  Google Scholar [3] P. Bolley, J. Camus and G. Métivier, Estimations de Schauder et régularité hölderienne pour une class de problémes aux limites singuliers, Comm. Partial Differential Equations, 11 (1986), 1135-1203. doi: 10.1080/03605308608820460.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995.  Google Scholar [6] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar [7] P. Daskalopoulos and K. Lee, Hölder regularity of solutions of degenerate elliptic and parabolic equations, J. Funct. Anal., 201 (2003), 341-379. doi: 10.1016/S0022-1236(02)00045-9.  Google Scholar [8] H. Dong and S. Kim, Partial Schauder estimates for second order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.  Google Scholar [9] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in Boundary Problems in Differential Equations, Univ. of Wisconsin Press, (1960), 97-120.  Google Scholar [10] G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems, Lecture Notes in Mathematics 8, Springer-Verlag, Berlin, 1965.  Google Scholar [11] G. Fichera, Existence theorems in elasticity, in Handbuch der Physik. Band VIa/2: Festkörpermechanik. II Springer-Verlag, Berlin, (1972), 347-389. Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of 1998 edition, Springer-Verlag, Berlin, 2001.  Google Scholar [13] J. Hong and G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications, Int. Math. Res. Not., 13 (2012), 2889-2941.  Google Scholar [14] M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain, Dokl. Akad. Nauk SSR, 77 (1951), 181-183 [Russian].  Google Scholar [15] N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Zh., 17 (1976), 290-303 [Russian]; English transl. in Sib. Math. J., 17 (1976), 226-236.  Google Scholar [16] 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 (1983), 67-98.  Google Scholar [17] N. V. Krylov, On estimates of the maximum of a solution of a parabolic equation and estimates of the distributions of a semimartingale, Mat. Sb., 130 (1986), 207-221 [Russian]; English transl. in Math. USSR-Sb., 58 (1987), 207-221.  Google Scholar [18] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/012.  Google Scholar [19] M. Langlais, On the continuous solutions of a degenerate elliptic equation, Proc. London Math. Soc., 50 (1985), 282-298. doi: 10.1112/plms/s3-50.2.282.  Google Scholar [20] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic partial differential equations of second order, J. Math. Anal. Appl., 113 (1986), 422-440. doi: 10.1016/0022-247X(86)90314-8.  Google Scholar [21] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [22] G. M. Lieberman, The maximum principle for equations with compositie coefficients, Electronic J. Differential Equations, 2000 (2000), 1-17.  Google Scholar [23] G. M. Lieberman, Solutions of singular elliptic equations via the oblique derivative problem, Ann. Univ. Ferrara, 57 (2011), 121-172. doi: 10.1007/s11565-010-0113-1.  Google Scholar [24] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [25] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for fully nonlinear elliptic equations, Trans. Amer. Math. Soc., 295 (1986), 509-546. doi: 10.1090/S0002-9947-1986-0833695-6.  Google Scholar [26] J. L. Lions, Sur les problèmes aux limites du type dérivée oblique, Ann. of Math. (2), 64 (1956), 207-239. doi: 10.2307/1969970.  Google Scholar [27] P. L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Bellman operator, Math. Z., 191 (1986), 1-15. doi: 10.1007/BF01163605.  Google Scholar [28] A. I. Nazarov, Hölder estimates for bounded solutions of problems with an oblique derivative for parabolic equations of nondivergence form, Probl. Math. Anal., 11 (1990), 37-46 [Russian]. English transl in J. Soviet Math. 64 (1993), 1247-1252.  Google Scholar [29] O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Translated from the Russian by Paul C. Fife, Plenum Press, New York, 1973.  Google Scholar [30] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl., 74 (1966), 15-30. doi: 10.1007/BF02416445.  Google Scholar [31] M. V. Safonov, On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 1272-1287 [Russian]; English transl. in Math. USSR-Izv. 33 (1989), 597-612.  Google Scholar [32] 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.  Google Scholar [33] L. Wang, A maximum principle for elliptic and parabolic equations with oblique derivative boundary conditions, J. Partial Differential Equations, 5 (1992), 23-27.  Google Scholar

show all references

##### References:
 [1] A. D. Aleksandrov, Certain estimates for the Dirichlet problem, Dokl Akad. Nauk SSSR, 134 (1961), 1001-1004 [Russian]; English transl. in Soviet Math. Dokl. 1 (1961), 1151-1154.  Google Scholar [2] I. Ya. Bakel'man, Theory of quasilinear elliptic equations, Sibirsk. Mat. Zh., 2 (1961), 179-186 [Russian].  Google Scholar [3] P. Bolley, J. Camus and G. Métivier, Estimations de Schauder et régularité hölderienne pour une class de problémes aux limites singuliers, Comm. Partial Differential Equations, 11 (1986), 1135-1203. doi: 10.1080/03605308608820460.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, RI, 1995.  Google Scholar [6] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-9982-5.  Google Scholar [7] P. Daskalopoulos and K. Lee, Hölder regularity of solutions of degenerate elliptic and parabolic equations, J. Funct. Anal., 201 (2003), 341-379. doi: 10.1016/S0022-1236(02)00045-9.  Google Scholar [8] H. Dong and S. Kim, Partial Schauder estimates for second order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.  Google Scholar [9] G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, in Boundary Problems in Differential Equations, Univ. of Wisconsin Press, (1960), 97-120.  Google Scholar [10] G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems, Lecture Notes in Mathematics 8, Springer-Verlag, Berlin, 1965.  Google Scholar [11] G. Fichera, Existence theorems in elasticity, in Handbuch der Physik. Band VIa/2: Festkörpermechanik. II Springer-Verlag, Berlin, (1972), 347-389. Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of 1998 edition, Springer-Verlag, Berlin, 2001.  Google Scholar [13] J. Hong and G. Huang, $L^p$ and Hölder estimates for a class of degenerate elliptic partial differential equations and its applications, Int. Math. Res. Not., 13 (2012), 2889-2941.  Google Scholar [14] M. V. Keldyš, On certain cases of degeneration of equations of elliptic type on the boundary of a domain, Dokl. Akad. Nauk SSR, 77 (1951), 181-183 [Russian].  Google Scholar [15] N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Zh., 17 (1976), 290-303 [Russian]; English transl. in Sib. Math. J., 17 (1976), 226-236.  Google Scholar [16] 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 (1983), 67-98.  Google Scholar [17] N. V. Krylov, On estimates of the maximum of a solution of a parabolic equation and estimates of the distributions of a semimartingale, Mat. Sb., 130 (1986), 207-221 [Russian]; English transl. in Math. USSR-Sb., 58 (1987), 207-221.  Google Scholar [18] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/012.  Google Scholar [19] M. Langlais, On the continuous solutions of a degenerate elliptic equation, Proc. London Math. Soc., 50 (1985), 282-298. doi: 10.1112/plms/s3-50.2.282.  Google Scholar [20] G. M. Lieberman, Mixed boundary value problems for elliptic and parabolic partial differential equations of second order, J. Math. Anal. Appl., 113 (1986), 422-440. doi: 10.1016/0022-247X(86)90314-8.  Google Scholar [21] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [22] G. M. Lieberman, The maximum principle for equations with compositie coefficients, Electronic J. Differential Equations, 2000 (2000), 1-17.  Google Scholar [23] G. M. Lieberman, Solutions of singular elliptic equations via the oblique derivative problem, Ann. Univ. Ferrara, 57 (2011), 121-172. doi: 10.1007/s11565-010-0113-1.  Google Scholar [24] G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific, Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar [25] G. M. Lieberman and N. S. Trudinger, Nonlinear oblique boundary value problems for fully nonlinear elliptic equations, Trans. Amer. Math. Soc., 295 (1986), 509-546. doi: 10.1090/S0002-9947-1986-0833695-6.  Google Scholar [26] J. L. Lions, Sur les problèmes aux limites du type dérivée oblique, Ann. of Math. (2), 64 (1956), 207-239. doi: 10.2307/1969970.  Google Scholar [27] P. L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Bellman operator, Math. Z., 191 (1986), 1-15. doi: 10.1007/BF01163605.  Google Scholar [28] A. I. Nazarov, Hölder estimates for bounded solutions of problems with an oblique derivative for parabolic equations of nondivergence form, Probl. Math. Anal., 11 (1990), 37-46 [Russian]. English transl in J. Soviet Math. 64 (1993), 1247-1252.  Google Scholar [29] O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form, Translated from the Russian by Paul C. Fife, Plenum Press, New York, 1973.  Google Scholar [30] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl., 74 (1966), 15-30. doi: 10.1007/BF02416445.  Google Scholar [31] M. V. Safonov, On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 1272-1287 [Russian]; English transl. in Math. USSR-Izv. 33 (1989), 597-612.  Google Scholar [32] 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.  Google Scholar [33] L. Wang, A maximum principle for elliptic and parabolic equations with oblique derivative boundary conditions, J. Partial Differential Equations, 5 (1992), 23-27.  Google Scholar
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