American Institute of Mathematical Sciences

Advanced
April  2021, 20(4): 1363-1383. doi: 10.3934/cpaa.2021024

$ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations

Arunima Bhattacharya 1, and  Micah Warren 2,,

1. 

Department of Mathematics, University of Washington, Seattle, WA 98195, USA
2. 

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

* Corresponding author

Received  September 2020 Revised  December 2020 Published  April 2021 Early access  March 2021

In this paper, we show explicit $ C^{2, \alpha} $ interior estimates for viscosity solutions of fully non-linear, uniformly elliptic equations, which are close to linear equations and we also compute an explicit bound for the closeness.

Keywords: Non-linear, uniformly elliptic, viscosity solutions, Cordes-Nirenberg, non concave/convex.
Mathematics Subject Classification: Primary: 35A09, 35B45; Secondary: 35B20.
Citation: Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1363-1383. doi: 10.3934/cpaa.2021024
References:
[1]

S. N. Armstrong, L. E. Silvestre and C. K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Commun. Pure Appl. Math., 65 (2012), 1169-1184.  doi: 10.1002/cpa.21394.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. A. Caffarelli and Y. Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J., 49 (2000), 681-695.  doi: 10.1512/iumj.2000.49.1901.  Google Scholar

[4]

X. Cabré and L. A. Caffarelli, Interior $C^{2, \alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  doi: 10.1016/S0021-7824(03)00029-1.  Google Scholar

[5]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.   Google Scholar

[6]

Y. Cao, D. S. Li and L. H. Wang, A priori estimates for classical solutions of fully nonlinear elliptic equations, Sci. China Math., 54 (2011), 457-462.  doi: 10.1007/s11425-010-4092-6.  Google Scholar

[7]

T. C. Collins, $C^{2, \alpha}$ estimates for nonlinear elliptic equations of twisted type, Calc. Var. Partial Differ. Equ., 55 (2016), 1-11.  doi: 10.1007/s00526-015-0950-y.  Google Scholar

[8]

H. O. Cordes, Über die erste randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen, Math. Ann., 131 (1956), 278-312.  doi: 10.1007/BF01342965.  Google Scholar

[9]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, in Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 1961.  Google Scholar

[10]

H. J. Dong, Recent progress in the $ l\_p $ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theor. Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021.  Google Scholar

[11]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[13]

Q. B. Huang, Regularity theory for ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 36 (2019), 1869-1902.  doi: 10.1016/j.anihpc.2019.06.001.  Google Scholar

[14]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.   Google Scholar

[15]

N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv., 16 (1981), 151-164.   Google Scholar

[16]

N. Nadirashvili and S. Vlăduţ, Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl., 89 (2008), 107-113.  doi: 10.1016/j.matpur.2007.10.004.  Google Scholar

[17]

N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.  Google Scholar

[18]

N. Nadirashvili and S. Vlăduţ, Singular solution to special lagrangian equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 1179-1188.  doi: 10.1016/j.anihpc.2010.05.001.  Google Scholar

[19]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Commun. Pure Appl. Math., 6 (1953), 103-156.  doi: 10.1002/cpa.3160060105.  Google Scholar

[20]

L. Nirenberg, On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations, in Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton, N. J., 1954.  Google Scholar

[21]

V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.   Google Scholar

[22]

O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[23]

J. Streets and M. Warren, Evans-Krylov estimates for a nonconvex Monge-Ampère equation, Math. Ann., 365 (2016), 805-834.  doi: 10.1007/s00208-015-1293-x.  Google Scholar

[24]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[25]

Y. Yuan, A priori estimates for solutions of fully nonlinear special Lagrangian equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 261-270.  doi: 10.1016/S0294-1449(00)00065-2.  Google Scholar

show all references

References:
[1]

S. N. Armstrong, L. E. Silvestre and C. K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Commun. Pure Appl. Math., 65 (2012), 1169-1184.  doi: 10.1002/cpa.21394.  Google Scholar

[2]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Soc., 1995. doi: 10.1090/coll/043.  Google Scholar

[3]

L. A. Caffarelli and Y. Yuan, A priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J., 49 (2000), 681-695.  doi: 10.1512/iumj.2000.49.1901.  Google Scholar

[4]

X. Cabré and L. A. Caffarelli, Interior $C^{2, \alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl., 82 (2003), 573-612.  doi: 10.1016/S0021-7824(03)00029-1.  Google Scholar

[5]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5 (1958), 105-126.   Google Scholar

[6]

Y. Cao, D. S. Li and L. H. Wang, A priori estimates for classical solutions of fully nonlinear elliptic equations, Sci. China Math., 54 (2011), 457-462.  doi: 10.1007/s11425-010-4092-6.  Google Scholar

[7]

T. C. Collins, $C^{2, \alpha}$ estimates for nonlinear elliptic equations of twisted type, Calc. Var. Partial Differ. Equ., 55 (2016), 1-11.  doi: 10.1007/s00526-015-0950-y.  Google Scholar

[8]

H. O. Cordes, Über die erste randwertaufgabe bei quasilinearen differentialgleichungen zweiter ordnung in mehr als zwei variablen, Math. Ann., 131 (1956), 278-312.  doi: 10.1007/BF01342965.  Google Scholar

[9]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, in Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 1961.  Google Scholar

[10]

H. J. Dong, Recent progress in the $ l\_p $ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theor. Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021.  Google Scholar

[11]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[13]

Q. B. Huang, Regularity theory for ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 36 (2019), 1869-1902.  doi: 10.1016/j.anihpc.2019.06.001.  Google Scholar

[14]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR-Izv., 20 (1983), 459-492.   Google Scholar

[15]

N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv., 16 (1981), 151-164.   Google Scholar

[16]

N. Nadirashvili and S. Vlăduţ, Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl., 89 (2008), 107-113.  doi: 10.1016/j.matpur.2007.10.004.  Google Scholar

[17]

N. Nadirashvili, V. Tkachev and S. Vlăduţ, Nonlinear Elliptic Equations and Nonassociative Algebras, American Mathematical Soc., 2014. doi: 10.1090/surv/200.  Google Scholar

[18]

N. Nadirashvili and S. Vlăduţ, Singular solution to special lagrangian equations, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 27 (2010), 1179-1188.  doi: 10.1016/j.anihpc.2010.05.001.  Google Scholar

[19]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Commun. Pure Appl. Math., 6 (1953), 103-156.  doi: 10.1002/cpa.3160060105.  Google Scholar

[20]

L. Nirenberg, On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations, in Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton, N. J., 1954.  Google Scholar

[21]

V. P. Pingali, $C^{2, \alpha}$estimates and existence results for a nonconcave PDE, Electron. J. Differ. Equ., 168 (2016), 1-10.   Google Scholar

[22]

O. Savin, Small perturbation solutions for elliptic equations, Commun Partial Differ. Equ., 3 (2007), 557-578.  doi: 10.1080/03605300500394405.  Google Scholar

[23]

J. Streets and M. Warren, Evans-Krylov estimates for a nonconvex Monge-Ampère equation, Math. Ann., 365 (2016), 805-834.  doi: 10.1007/s00208-015-1293-x.  Google Scholar

[24]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[25]

Y. Yuan, A priori estimates for solutions of fully nonlinear special Lagrangian equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 261-270.  doi: 10.1016/S0294-1449(00)00065-2.  Google Scholar

[1]

Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481
[2]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i
[3]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165
[4]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[5]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
[6]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[7]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[8]

Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021131
[9]

Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155
[10]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[11]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535
[12]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[13]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004
[14]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[15]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[16]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193
[17]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[18]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032
[19]

Tarek Saanouni. Non-linear bi-harmonic Choquard equations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221
[20]

Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021014

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (67)
  • HTML views (121)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences