We consider averages $\kappa$ of spectral measures of rank one
perturbations with respect to a sigma-finite measure $\nu$. It is shownhow various degrees of continuity of $\nu$ with respect to Hausdorff measures are inherited by $\kappa$. This extends Kotani's trick where $\nu$ is simply the Lebesgue measure.

In this talk,
we study weighted $L^p$-norm inequalities for general spectral multipliers
for self-adjoint positive definite operators on $L^2(X)$, where $X$ is
a space of homogeneous type. We show that the sharp weighted H\"ormander-type
spectral multiplier theorems follow from the appropriate estimates of the $L^2$
norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding
heat kernels. These results are applicable to spectral multipliers for group
invariant Laplace operators acting on Lie groups of polynomial growth and elliptic
operators on compact manifolds.

The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersive wave equation which was proposed to study the stability of one soliton solution of the KdV equation under the influence of weak transversal perturbations. It is well know that some closed-form solutions can be obtained by function which have a Wronskian determinant form. It is of interest to study KP with an arbitrary initial condition and see whether the solution converges to any closed-form solution asymptotically. To reveal the answer to this question both numerically and theoretically, we consider different types of initial conditions, including one-line soliton, V-shape wave and cross-shape wave, and investigate the behavior of solutions asymptotically. We provides a detail description of classification on the results.

The challenge of numerical approach comes from the unbounded domain and unvanished solutions in the infinity. In order to do numerical computation on the finite domain, boundary conditions need to be imposed carefully. Due to the non-periodic boundary conditions, the standard spectral method with Fourier methods involving trigonometric polynomials cannot be used. We proposed a new spectral method with a window technique which will make the boundary condition periodic and allow the usage of the classical approach. We demonstrate the robustness and efficiency of our methods through numerous simulations.

In this work we present and efficient approach to homogenization for a class of static Hamilton-Jacobi (HJ) equations, which we call metric HJ equations. We relate the solutions of the HJ equations to the distance function in a corresponding Riemannian or Finslerian metric. The metric approach allows us to conclude that the homogenized equation also induces a metric. The advantage of the method is that we can solve just one auxiliary equation to recover the homogenized Hamiltonian. This is significant improvement over existing methods which require the solution of the cell problem (or a variational problem) for each value of p. Computational results are presented and compared with analytic results when available for piece-wise constant periodic and random speed functions.

We will also discuss some recent results on homogenization of second order fully nonlinear equations.