# American Institute of Mathematical Sciences

September  2012, 17(6): 1693-1706. doi: 10.3934/dcdsb.2012.17.1693

## The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions

 1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois, 60660, United States, United States, United States

Received  July 2011 Revised  December 2011 Published  May 2012

Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then used in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations.
Citation: Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
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##### References:
 [1] Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 [2] Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461 [3] Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 [4] María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207 [5] Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441 [6] Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389 [7] Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793 [8] Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649 [9] Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493 [10] Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 [11] Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 [12] Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 [13] Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623 [14] Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331 [15] Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176 [16] Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $BV$ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405 [17] Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 [18] Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 [19] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [20] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

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