# American Institute of Mathematical Sciences

• Previous Article
Justification of leading order quasicontinuum approximations of strongly nonlinear lattices
• DCDS Home
• This Issue
• Next Article
A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity
September  2014, 34(9): 3383-3402. doi: 10.3934/dcds.2014.34.3383

## On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions

 1 Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui Province, China

Received  August 2013 Revised  December 2013 Published  March 2014

Spacetime convexity is a basic geometric property of the solutions of parabolic equations. In this paper, we study microscopic convexity properties of spacetime convex solutions of fully nonlinear parabolic partial differential equations and give a new simple proof of the constant rank theorem in [11].
Citation: 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
##### References:

show all references

##### References:
 [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] 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 [3] 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 [4] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [5] 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 [6] 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 [7] 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 [8] 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 [9] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395 [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, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2019 Impact Factor: 1.338