American Institute of Mathematical Sciences

Advanced
April  2002, 8(2): 331-359. doi: 10.3934/dcds.2002.8.331

Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations

Xavier Cabré 1,

1. 

Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Av. Diagonal 647. 08028 Barcelona, Spain

Revised  November 2001 Published  January 2002

In these notes we describe the Alexandroff-Bakelman-Pucci estimate and the Krylov-Safonov Harnack inequality for solutions of $Lu = f(x)$, where $L$ is a second order uniformly elliptic operator in nondivergence form with bounded measurable coefficients. It is the purpose of these notes to present several applications of these inequalities to the study of nonlinear elliptic equations.
The first topic is the maximum principle for the operator $L$, and its applications to the moving planes method and to symmetry properties of positive solutions of semilinear problems. The second topic is a short introduction to the regularity theory for solutions of fully nonlinear elliptic equations. We prove a $C^{1,\alpha}$ estimate for classical solutions, we introduce the notion of viscosity solution, and we study Jensen’s approximate solutions.
Keywords: maximum principles, Nonlinear elliptic PDE, fully nonlinear equations., a priori estimates, symmetry properties.
Mathematics Subject Classification: 35J60, 35B5.
Citation: Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331
[1]

Wenmin Sun, Jiguang Bao. New maximum principles for fully nonlinear ODEs of second order. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 813-823. doi: 10.3934/dcds.2007.19.813
[2]

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
[3]

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
[4]

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
[5]

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
[6]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[7]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[8]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[9]

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
[10]

Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
[11]

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
[12]

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
[13]

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
[14]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[15]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709
[16]

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
[17]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[18]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343
[19]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[20]

Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences