Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$\Delta u = -\chi_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss

Self-propagating High temperature Synthesis (SHS) in the high activation energy regime

We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit turns out to be the Stefan problem for supercooled water with spatially inhomogeneous coefficients. Although the present paper leaves open mathematical questions concerning the convergence, our precise form of the limit problem suggest a strikingly simple explanation for the numerically observed pulsating waves

Decay of correlations in the dissipative two-state system

We study the equilibrium correlation function of the polaron-dressed tunnelling operator in the dissipative two-state system and compare the asymptoptic dynamics with that of the position correlations. For an Ohmic spectral density with the damping strength $K=1/2$, the correlation functions are obtained in analytic form for all times at any $T$ and any bias. For $K<1$, the asymptotic dynamics is found by using a diagrammatic approach within a Coulomb gas representation. At T=0, the tunnelling or coherence correlations drop as $t^{-2K}$, whereas the position correlations show universal decay $\propto t^{-2}$. The former decay law is a signature of unscreened attractive charge-charge interactions, while the latter is due to unscreened dipole-dipole interactions.Comment: 5 pages, 5 figures, to be published in Europhys. Let