# American Institute of Mathematical Sciences

June  2011, 31(2): 385-406. doi: 10.3934/dcds.2011.31.385

## On $C^0$-variational solutions for Hamilton-Jacobi equations

 1 Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova, Italy, Italy

Received  March 2010 Revised  April 2011 Published  June 2011

For evolutive Hamilton-Jacobi equations, we propose a refined definition of $C^0$-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the semigroup property for these solutions. In the case of $p$-convex Hamiltonians, when variational solutions are known to be identical to viscosity solutions, we verify directly the semigroup property by using minmax techniques. In the non-convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the separation of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions.
Citation: Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
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