  We study the Cauchy problem for the Hamilton-Jacobi equation with a
semiconcave initial condition. We prove an inequality between two types of weak
solutions emanating from such an initial condition (the variational and the
viscosity solution).We also give conditions for an explicit semi-concave
function to be a viscosity solution. These conditions generalize the entropy
inequality characterizing piecewise smooth solutions of scalar conservation
laws in dimension one.
