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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.

[2]

I. Ya. Bakel'man, Theory of quasilinear elliptic equations,, Sibirsk. Mat. Zh., 2 (1961), 179.

[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. doi: 10.1080/03605308608820460.

[4]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math. (2), 130 (1989), 189. doi: 10.2307/1971480.

[5]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations,, American Mathematical Society, (1995).

[6]

M. Chipot, Elliptic Equations: An Introductory Course,, Birkhäuser, (2009). doi: 10.1007/978-3-7643-9982-5.

[7]

P. Daskalopoulos and K. Lee, Hölder regularity of solutions of degenerate elliptic and parabolic equations,, J. Funct. Anal., 201 (2003), 341. doi: 10.1016/S0022-1236(02)00045-9.

[8]

H. Dong and S. Kim, Partial Schauder estimates for second order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481. doi: 10.1007/s00526-010-0348-9.

[9]

G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, in Boundary Problems in Differential Equations, (1960), 97.

[10]

G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems,, Lecture Notes in Mathematics 8, (1965).

[11]

G. Fichera, Existence theorems in elasticity,, in Handbuch der Physik. Band VIa/2: Festkörpermechanik. II Springer-Verlag, (1972), 347.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of 1998 edition, (1998).

[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.

[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.

[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.

[16]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, Izv. Akad. Nauk SSSR, 47 (1983), 75.

[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.

[18]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces,, American Mathematical Society, (1996). doi: 10.1090/gsm/012.

[19]

M. Langlais, On the continuous solutions of a degenerate elliptic equation,, Proc. London Math. Soc., 50 (1985), 282. doi: 10.1112/plms/s3-50.2.282.

[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. doi: 10.1016/0022-247X(86)90314-8.

[21]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996). doi: 10.1142/3302.

[22]

G. M. Lieberman, The maximum principle for equations with compositie coefficients,, Electronic J. Differential Equations, 2000 (2000), 1.

[23]

G. M. Lieberman, Solutions of singular elliptic equations via the oblique derivative problem,, Ann. Univ. Ferrara, 57 (2011), 121. doi: 10.1007/s11565-010-0113-1.

[24]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations,, World Scientific, (2013). doi: 10.1142/8679.

[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. doi: 10.1090/S0002-9947-1986-0833695-6.

[26]

J. L. Lions, Sur les problèmes aux limites du type dérivée oblique,, Ann. of Math. (2), 64 (1956), 207. doi: 10.2307/1969970.

[27]

P. L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Bellman operator,, Math. Z., 191 (1986), 1. doi: 10.1007/BF01163605.

[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.

[29]

O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form,, Translated from the Russian by Paul C. Fife, (1973).

[30]

C. Pucci, Limitazioni per soluzioni di equazioni ellittiche,, Ann. Mat. Pura Appl., 74 (1966), 15. doi: 10.1007/BF02416445.

[31]

M. V. Safonov, On the classical solution of nonlinear elliptic equations of second order,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 1272.

[32]

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.

[33]

L. Wang, A maximum principle for elliptic and parabolic equations with oblique derivative boundary conditions,, J. Partial Differential Equations, 5 (1992), 23.

show all references

References:
[1]

A. D. Aleksandrov, Certain estimates for the Dirichlet problem,, Dokl Akad. Nauk SSSR, 134 (1961), 1001.

[2]

I. Ya. Bakel'man, Theory of quasilinear elliptic equations,, Sibirsk. Mat. Zh., 2 (1961), 179.

[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. doi: 10.1080/03605308608820460.

[4]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math. (2), 130 (1989), 189. doi: 10.2307/1971480.

[5]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations,, American Mathematical Society, (1995).

[6]

M. Chipot, Elliptic Equations: An Introductory Course,, Birkhäuser, (2009). doi: 10.1007/978-3-7643-9982-5.

[7]

P. Daskalopoulos and K. Lee, Hölder regularity of solutions of degenerate elliptic and parabolic equations,, J. Funct. Anal., 201 (2003), 341. doi: 10.1016/S0022-1236(02)00045-9.

[8]

H. Dong and S. Kim, Partial Schauder estimates for second order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481. doi: 10.1007/s00526-010-0348-9.

[9]

G. Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order,, in Boundary Problems in Differential Equations, (1960), 97.

[10]

G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems,, Lecture Notes in Mathematics 8, (1965).

[11]

G. Fichera, Existence theorems in elasticity,, in Handbuch der Physik. Band VIa/2: Festkörpermechanik. II Springer-Verlag, (1972), 347.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of 1998 edition, (1998).

[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.

[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.

[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.

[16]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain,, Izv. Akad. Nauk SSSR, 47 (1983), 75.

[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.

[18]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces,, American Mathematical Society, (1996). doi: 10.1090/gsm/012.

[19]

M. Langlais, On the continuous solutions of a degenerate elliptic equation,, Proc. London Math. Soc., 50 (1985), 282. doi: 10.1112/plms/s3-50.2.282.

[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. doi: 10.1016/0022-247X(86)90314-8.

[21]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996). doi: 10.1142/3302.

[22]

G. M. Lieberman, The maximum principle for equations with compositie coefficients,, Electronic J. Differential Equations, 2000 (2000), 1.

[23]

G. M. Lieberman, Solutions of singular elliptic equations via the oblique derivative problem,, Ann. Univ. Ferrara, 57 (2011), 121. doi: 10.1007/s11565-010-0113-1.

[24]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations,, World Scientific, (2013). doi: 10.1142/8679.

[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. doi: 10.1090/S0002-9947-1986-0833695-6.

[26]

J. L. Lions, Sur les problèmes aux limites du type dérivée oblique,, Ann. of Math. (2), 64 (1956), 207. doi: 10.2307/1969970.

[27]

P. L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Bellman operator,, Math. Z., 191 (1986), 1. doi: 10.1007/BF01163605.

[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.

[29]

O. A. Oleĭnik and E. V. Radkevič, Second Order Equations with Nonnegative Characteristic Form,, Translated from the Russian by Paul C. Fife, (1973).

[30]

C. Pucci, Limitazioni per soluzioni di equazioni ellittiche,, Ann. Mat. Pura Appl., 74 (1966), 15. doi: 10.1007/BF02416445.

[31]

M. V. Safonov, On the classical solution of nonlinear elliptic equations of second order,, Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988), 1272.

[32]

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.

[33]

L. Wang, A maximum principle for elliptic and parabolic equations with oblique derivative boundary conditions,, J. Partial Differential Equations, 5 (1992), 23.

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