A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.
Sharp constants are exhibited in exponential inequalities corresponding to the limiting case of the Sobolev inequalities in Lorentz-Sobolev spaces of arbitrary order.
We investigate finite difference schemes which approximate 2 × 2 one-dimensional
linear dissipative hyperbolic systems. We show that it is possible to introduce some suitable modifications
in standard upwinding schemes, which keep into account the long-time behavior of the
solutions, to yield numerical approximations which are increasingly accurate for large times when
computing small perturbations of stable asymptotic states, respectively, around stationary solutions
and in the diffusion (Chapman-Enskog) limit.
Numericalmethods for Volterra integral equations with discontinuous kernel need to be tuned to their peculiar form. Here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, we first investigate about the behavior of the solution of a suitable (basic) test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem.
Volterra integral equations
Trapezoidal method
Direct quadrature methods
Discontinuous kernel
Constant delay
