Solving Elliptic equations with sharp-edged interfaces is a challenging problem for most existing methods, especially when the solution is highly oscillatory. Nonetheless, it has wide applications in engineering and science. In the first part of this talk, I will present a non-traditional finite element method for solving matrix coefficient elliptic equations with shape-edged interfaces with 2nd order accuracy in L-infinity norm. In the second part of this talk, I will switch to a topics on interface shape classification, where the goal is to characterize a shape using the least amount of data, to generate a shape library and to match a given shape to one of the shapes in the library after certain rotation, scaling and shifting. I will present a novel method by using the response matrix data, motivated by a direct imaging method for inverse problems.

In this talk we discuss Bernstein estimates for weakly fully
non-linear elliptic systems. We are particularly
interested in systems that arise in the stochastic optimal control
problems of hybrid systems. For these a
generalization of Bernstein estimates for first and second derivatives
of classical solutions will be presented.

We consider random Dirac operators on a strip of width 2L of the form $J\partial+V$ where J is the $2L \times 2L $ symplectic form and V a hermitian matrix-valued random potential satisfying a time reversal symmetry property.
The operator can be analyzed using transfer matrices. The time reversal symmetry forces the transfer matrices to be in the group $SO^*(2L)$. This leads to symmetry and Kramer's degeneracy for the Lyapunov spectrum which forces two Lyapunov exponents to be zero if L is odd. Adopting a criterion
by Goldsheid and Margulis one proves that these are the only vanishing Lyapunov exponents under sufficient randomness. Adopting Kotani theory one obtains a.c. spectrum of multiplicity two on the whole real line. If moreover the random potential includes i.i.d., a.c. distributed matrix Diracpeaks on a lattice in $\RR$, we can adopt the work of Jaksic and Last to prove that the a.c. spectrum is pure. This is a big contrast to the case where L is even and no Lyapunov exponent vanishes for sufficient randomness. There one expects to get pure
point spectrum using similar techniques as in the one dimensional Anderson model. (joint work with H. Schulz-Baldes)