American Institute of Mathematical Sciences

Advanced
June  2010, 28(2): 789-807. doi: 10.3934/dcds.2010.28.789

A structural condition for microscopic convexity principle

Baojun Bian 1, and  Pengfei Guan 2,

1. 

Department of mathematics, Tongji University, Shanghai 200092, China
2. 

Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

Received  February 2010 Revised  April 2010 Published  April 2010

The arguments in paper [2] have been refined to prove amicroscopic convexity principle for fully nonlinear ellipticequation under a more natural structure condition. We also consider the correspondingresult for the partial convexity case.
Keywords: Convexity, partial convexity, constant rank principle., fully nonlinear equations.
Mathematics Subject Classification: Primary: 35K10, 35B9.
Citation: Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789
[1]

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
[2]

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
[3]

Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009
[4]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
[5]

David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
[6]

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
[7]

Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007
[8]

Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021
[9]

Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
[10]

Christopher Goodrich, Carlos Lizama. Positivity, monotonicity, and convexity for convolution operators. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4961-4983. doi: 10.3934/dcds.2020207
[11]

Arrigo Cellina, Carlo Mariconda, Giulia Treu. Comparison results without strict convexity. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 57-65. doi: 10.3934/dcdsb.2009.11.57
[12]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[13]

Victor Isakov. On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions. Inverse Problems & Imaging, 2019, 13 (5) : 983-1006. doi: 10.3934/ipi.2019044
[14]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[15]

Chadi Nour, Ron J. Stern, Jean Takche. Generalized exterior sphere conditions and $\varphi$-convexity. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 615-622. doi: 10.3934/dcds.2011.29.615
[16]

Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109
[17]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[18]

Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005
[19]

Kazuhiro Ishige, Paolo Salani. On a new kind of convexity for solutions of parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 851-864. doi: 10.3934/dcdss.2011.4.851
[20]

Martin Gugat, Rüdiger Schultz, Michael Schuster. Convexity and starshapedness of feasible sets in stationary flow networks. Networks & Heterogeneous Media, 2020, 15 (2) : 171-195. doi: 10.3934/nhm.2020008

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy