American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation
  • CPAA Home
  • This Issue
  • Next Article
    Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity
March  2021, 20(3): 1241-1261. doi: 10.3934/cpaa.2021019

Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

Lidan Wang 1, , Lihe Wang 1,2, and  Chunqin Zhou 3,,

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
3. 

School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author

Received  August 2020 Revised  December 2020 Published  March 2021 Early access  February 2021

Fund Project: The third author is partially supported by NSFC Grant 11771285 and 12031012

In this paper, we consider the positive viscosity solutions for certain fully nonlinear uniformly elliptic equations in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle, we classify all positive solutions as three categories in unbounded cylinder. Two special solution spaces (exponential growth at one end and exponential decay at the another) are one dimensional, independently, while solutions in the third solution space can be controlled by the solutions in the other two special solution spaces under some conditions, respectively.

Keywords: Fully nonlinear elliptic equations, unbounded cylinder, Aleksandrov-Bakelman-Pucci maximum principle, classification of solutions.
Mathematics Subject Classification: Primary: 35J60.
Citation: Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
References:
[1] S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of n-body schrodinger operations, Princeton, New Jersey: Princeton University Press, 1982.   Google Scholar
[2]

B. Avelin and V. Julin, A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term, J. Funct. Anal., 272 (2017), 3176 – 3215. doi: 10.1016/j.jfa.2016.12.026.  Google Scholar

[3]

M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., 2008 (2008), 1-20.  doi: 10.1155/2008/178534.  Google Scholar

[4]

J. Bao, L. Wang and C. Zhou, Positive solutions to elliptic equations in unbounded cylinder, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1389-1400.  doi: 10.3934/dcdsb.2016001.  Google Scholar

[5]

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

[6]

L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V.  Google Scholar

[7]

L. Cafarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981) 621–640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar

[8]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[9]

I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo, The Aleksandrof Backelman Pucci weak maximum priciple for Fully nonlinear equtions in unbounded domains, Commun. Partial Differ. Equ., 30 (2005), 1863-1881.  doi: 10.1080/03605300500300030.  Google Scholar

[10]

S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166.  doi: 10.1112/blms/22.2.163.  Google Scholar

[11]

M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181.  doi: 10.1007/s11118-009-9129-5.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[13]

E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb., 126 (1985), 133-139.   Google Scholar

[14]

R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.  doi: 10.2307/1990054.  Google Scholar

[15]

M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izv. Akad. Nauk. Ser. Mat., 27 (1963), 45-60.   Google Scholar

[16]

A. Swiech, W1, p interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differ. Equ., 2 (1997), 1005-1027.   Google Scholar

[17]

P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are $C^{1, \alpha}$, Commun. Pure Appl. Math., 53 (2000), 799-810.  doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[18]

L. Wang, L. Wang and C. Zhou, The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders, J. Korean Math. Soc., 57 (2020), 1573-1590.  doi: 10.4134/JKMS.j190836.  Google Scholar

show all references

References:
[1] S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of n-body schrodinger operations, Princeton, New Jersey: Princeton University Press, 1982.   Google Scholar
[2]

B. Avelin and V. Julin, A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term, J. Funct. Anal., 272 (2017), 3176 – 3215. doi: 10.1016/j.jfa.2016.12.026.  Google Scholar

[3]

M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., 2008 (2008), 1-20.  doi: 10.1155/2008/178534.  Google Scholar

[4]

J. Bao, L. Wang and C. Zhou, Positive solutions to elliptic equations in unbounded cylinder, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1389-1400.  doi: 10.3934/dcdsb.2016001.  Google Scholar

[5]

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

[6]

L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V.  Google Scholar

[7]

L. Cafarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981) 621–640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar

[8]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[9]

I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo, The Aleksandrof Backelman Pucci weak maximum priciple for Fully nonlinear equtions in unbounded domains, Commun. Partial Differ. Equ., 30 (2005), 1863-1881.  doi: 10.1080/03605300500300030.  Google Scholar

[10]

S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166.  doi: 10.1112/blms/22.2.163.  Google Scholar

[11]

M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181.  doi: 10.1007/s11118-009-9129-5.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[13]

E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb., 126 (1985), 133-139.   Google Scholar

[14]

R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.  doi: 10.2307/1990054.  Google Scholar

[15]

M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izv. Akad. Nauk. Ser. Mat., 27 (1963), 45-60.   Google Scholar

[16]

A. Swiech, W1, p interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differ. Equ., 2 (1997), 1005-1027.   Google Scholar

[17]

P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are $C^{1, \alpha}$, Commun. Pure Appl. Math., 53 (2000), 799-810.  doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[18]

L. Wang, L. Wang and C. Zhou, The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders, J. Korean Math. Soc., 57 (2020), 1573-1590.  doi: 10.4134/JKMS.j190836.  Google Scholar

[1]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[2]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[3]

Jun Bao, Lihe Wang, Chunqin Zhou. Positive solutions to elliptic equations in unbounded cylinder. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1389-1400. doi: 10.3934/dcdsb.2016001
[4]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
[5]

Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200
[6]

Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439
[7]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[8]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[9]

Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007
[10]

Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383
[11]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004
[12]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[13]

Haiyun Deng, Hairong Liu, Long Tian. Classification of singular sets of solutions to elliptic equations. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2949-2964. doi: 10.3934/cpaa.2020129
[14]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[15]

Mustafa Hasanbulli, Yuri V. Rogovchenko. Classification of nonoscillatory solutions of nonlinear neutral differential equations. Conference Publications, 2009, 2009 (Special) : 340-348. doi: 10.3934/proc.2009.2009.340
[16]

Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307
[17]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[18]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[19]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[20]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (125)
  • HTML views (108)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences