
Previous Article
Rigidity of partially hyperbolic actions of property (T) groups
 DCDS Home
 This Issue

Next Article
Interaction estimates and Glimm functional for general hyperbolic systems
Discontinuous solutions for HamiltonJacobi equations: Uniqueness and regularity
1.  Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 602082730 
2.  Department of Mathematics, University of WisconsinMadison, Madison, WI 53706 
(*)$ \qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \equiv \lim$inf$_{y\rightarrow x, y\in\mathbb R^d\backslash\Gamma}\varphi(y).
The regularity of discontinuous solutions to HamiltonJacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The $L^1$accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bilateral solutions, $L$solutions, minimax solutions, and $L^\infty$solutions is also clarified.
[1] 
Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of HamiltonJacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793812. doi: 10.3934/cpaa.2006.5.793 
[2] 
Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for HamiltonJacobi equations. Communications on Pure and Applied Analysis, 2003, 2 (4) : 447459. doi: 10.3934/cpaa.2003.2.447 
[3] 
Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of HamiltonJacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389410. doi: 10.3934/cpaa.2007.6.389 
[4] 
Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of HamiltonJacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 16491659. doi: 10.3934/dcds.2016.36.1649 
[5] 
Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic HamiltonJacobi equations. Discrete and Continuous Dynamical Systems  B, 2010, 13 (3) : 647664. doi: 10.3934/dcdsb.2010.13.647 
[6] 
Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type HamiltonJacobi equations I: Cauchy problem. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 43454358. doi: 10.3934/dcds.2019176 
[7] 
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor SánchezMorgado. Physical solutions of the HamiltonJacobi equation. Discrete and Continuous Dynamical Systems  B, 2005, 5 (3) : 513528. doi: 10.3934/dcdsb.2005.5.513 
[8] 
Olga Bernardi, Franco Cardin. On $C^0$variational solutions for HamiltonJacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385406. doi: 10.3934/dcds.2011.31.385 
[9] 
David McCaffrey. A representational formula for variational solutions to HamiltonJacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 12051215. doi: 10.3934/cpaa.2012.11.1205 
[10] 
Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate HamiltonJacobi equations with OrnsteinUhlenbeck operator. Communications on Pure and Applied Analysis, 2019, 18 (3) : 9991021. doi: 10.3934/cpaa.2019049 
[11] 
Xia Li. Longtime asymptotic solutions of convex hamiltonjacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 51515162. doi: 10.3934/dcds.2017223 
[12] 
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763800. doi: 10.3934/dcds.2008.21.763 
[13] 
Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete and Continuous Dynamical Systems  B, 2019, 24 (7) : 31953210. doi: 10.3934/dcdsb.2018340 
[14] 
Eddaly Guerra, Héctor SánchezMorgado. Vanishing viscosity limits for spacetime periodic HamiltonJacobi equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 331346. doi: 10.3934/cpaa.2014.13.331 
[15] 
Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p$solutions of the stochastic NavierStokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations and Control Theory, 2017, 6 (3) : 409425. doi: 10.3934/eect.2017021 
[16] 
Anya Désilles, Hélène Frankowska. Explicit construction of solutions to the Burgers equation with discontinuous initialboundary conditions. Networks and Heterogeneous Media, 2013, 8 (3) : 727744. doi: 10.3934/nhm.2013.8.727 
[17] 
Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 18. doi: 10.3934/cpaa.2005.4.1 
[18] 
ZhiQiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initialboundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure and Applied Analysis, 2015, 14 (3) : 759792. doi: 10.3934/cpaa.2015.14.759 
[19] 
Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent HamiltonJacobi equations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 42254239. doi: 10.3934/dcds.2015.35.4225 
[20] 
Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2019, 18 (3) : 10911115. doi: 10.3934/cpaa.2019053 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]