Gibson 1st Floor SR

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We study nonnegative radial solutions to the problem

\begin{equation*}

\left\{

\begin{split}

-\Delta u = \lambda K(\left|x \right|) f(u), \quad x \in \Omega

\\u = 0 \quad \qquad \quad \qquad \mbox{if } \left|x \right| = r_0

\\u \rightarrow0 \quad \qquad \quad \qquad \mbox{as } \left|x \right|\rightarrow\infty,

\end{split} \right.

\end{equation*}

where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$,

$\Omega=\{x\in\ \mathbb{R}^{n}; n \textgreater 2, \left|x \right| \textgreater r_0\}$ and $K$ belongs to a class of functions such that $\lim_{r\rightarrow \infty}K(r)=0$. For classes of nonlinearities $f$ that are negative at the origin and sublinear at $\infty$ we discuss existence and uniqueness results when $\lambda$ is large.

\]

The energy of a deformed crystal is calculated in the context of a central force lattice model in two dimensions. When the crystal shape is a lattice polygon, it is shown that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for an infinite-range potential it is asymptotically valid as the lattice parameter tends to zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. A new bond counting approach is used, which reduces the problem to certain lattice point problems of number theory. The approach is then extended to more general convex regions with possibly curved boundary. The resulting surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differentiable. The method also yields an explicit interfacial energy for twin and phase boundaries.

Abstract: In this talk, I will discuss the proportionality between tree amplitudes and the ultraviolet divergences in their one-loop corrections in Yang-Mills and (N < 4) Super Yang-Mills theories in four-dimensions. From the point of view of local perturbative quantum field theory, i.e. Feynman diagrams, this proportionality is straightforward: ultraviolet divergences at loop-level are absorbed into coefficients of local operators/interaction vertices in the original tree-amplitude. Ultraviolet divergences in loop amplitudes are also calculable through on-shell methods. These methods ensure manifest gauge-invariance, even at loop-level (no ghosts), at the expense of manifest locality. From an on-shell perspective, the proportionality between the ultraviolet divergences the tree amplitudes is thus not guaranteed. I describe systematic structures which ensure proportionality, and their possible connections to other recent developments in the field.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also

be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.

Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena.

We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces.