Nonlinear Boundary Value Problems via Minimization on Orlicz-Sobolev Spaces

We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation \displaystyle -\mbox{div} (\phi(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} \Omega under Dirichlet boundary conditions, where $\Omega \subset {\bf R}^{N}$ is a bounded smooth domain, $\phi : (0,\infty)\longrightarrow (0,\infty)$ is a suitable continuous function and $f: \Omega \times {\bf R} \to {\bf R}$ satisfies the Carath\'eodory conditions, while $h$ is a measure.Comment: 14 page

Nagel scaling and relaxation in the kinetic Ising model on a n-isotopic chain

The kinetic Ising model on a n-isotopic chain is considered in the framework of Glauber dynamics. The chain is composed of N segments with n sites, each one occupied by a different isotope. Due to the isotopic mass difference, the n spins in each segment have different relaxation times in the absence of the interactions, and consequently the dynamics of the system is governed by multiple relaxation mechanisms. The solution is obtained in closed form for arbitrary n, by reducing the problem to a set of n coupled equations, and it is shown rigorously that the critical exponent z is equal to 2. Explicit results are obtained numerically for any temperature and it is also shown that the dynamic susceptibility satisfies the new scaling (Nagel scaling) proposed for glass-forming liquids. This is in agreement with our recent results (L. L. Goncalves, M. Lopez de Haro, J. Taguena-Martinez and R. B. Stinchcombe, Phys. Rev. Lett. 84, 1507 (2000)), which relate this new scaling function to multiple relaxation processes.Comment: 4 pages, 2 figures, presented at Ising Centennial Colloquium, to be published in the Proceedings (Brazilian Journal of Physics.