Rouben Rostamian

The principle of linearized stability for a class of degenerate diffusion equations

Here u0 is a given bounded nonnegative function, the functions b(s) and ,f(s), defined for s 2 0, are smooth for s > 0, f(0) > 0, /I(O) = 0, /Y(O) = + co, /I’(s) > 0 for s>O and fib-‘(~) is Lipschitz continuous for s 30 (the precise hypotheses on the data can be found in Section 2). Since p’(O) = co, the diffusion is degenerate near points where u = 0. In this paper we study the stability of steady-state solutions of Problem I by linearizing (1.1) near an equilibrium V. So let 0 E C*(D) satisfy

On the uniformization of the solutions of the porous medium equation inRN

The asymptotics of the solution of the porous medium equation are related to the size of the initial data measured in an optimal way. The universality of the separable solutions is established. Finally an interesting difference with the heat equation is pointed out.

Gradient estimates for degenerate diffusion equations II

We establish upper and lower bounds for various norms of solutions and their gradients for the equation u t = div (|∇ u | m−1 ∇ u ) in ℝ N in terms of the norms of the initial data. Based on the L ∞ estimate of ∇ u , we conclude that u ( x , t ) is Lipschitz continuous in space-time, for all t >0, whenever u ( x ,0) is in L 1 (ℝ N ).

Reflection and refraction of elastic waves for stratified materials

Abstract Formulas are derived for the reflection and transmission tensors associated with a plane elastic wave impinging obliquely upon a stratified slab interposed between two homogenous half-spaces. These tensors transform the components of the incident displacement field into the components of reflected and transmitted displacement fields. Both solid-solid and solid-liquid interfaces are considered.

Internal constraints in linear elasticity

Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyls decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.Almost all the results here apply to materials both with or without internal constraints. For internally constrained materials however, the verification of certain hypothesis is surprisingly non-trivial as indicated by the computation in the appendix.

HOMOGENIZATION, SYMMETRY, AND PERIODIZATION IN DIFFUSIVE RANDOM MEDIA

Abstract We present a systematic study of homogenization of diffusion in random media with emphasis on tile-based random microstructures. We give detailed examples of several such media starting from their physical descriptions, then construct the associated probability spaces and verify their ergodicity. After a discussion of material symmetries of random media, we derive criteria for the isotropy of the homogenized limits in tile-based structures. Furthermore, we study the periodization algorithm for the numerical approximation of the homogenized diffusion tensor and study the algorithms rate of convergence. For one dimensional tile-based media, we prove a central limit result, giving a concrete rate of convergence for periodization. We also provide numerical evidence for a similar central limit behavior in the case of two dimensional tile-based structures.

THEORETICAL ANALYSIS OF STABILITY OF AXIALLY SYMMETRIC ROTATING OBJECTS WITH REGARD TO ORIENTING APPLES

Inspection using machine vision offers the potential for improved food safety and quality. However, effectiveness of fruit inspection has been limited by the difficulty of appropriately orienting fruit for imaging. Commercial orientation systems have had limited impact due to mechanical complexity, cost, error, or some combination thereof. Preliminary tests demonstrated that apples could be oriented by rolling them down a track consisting of two parallel rails. After achieving sufficient angular velocity, the apples moved to an orientation where the stem/calyx axis was perpendicular to the direction of travel and parallel to the plane of the track. The purpose of the current study was to examine whether this orientation phenomenon could be explained in terms of the inertial characteristics of axially symmetric objects. Rotation of a free body around an axis of axial symmetry was found to be stable, while rotation around an axis perpendicular to this symmetric axis was not. Furthermore, comparisons of action values for two different mathematical models of apples indicate that inertial characteristics can be used to orient apples. The critical assumption for this analysis was that for the same object traveling the same path with different initial orientations, lower action values represent preferred motions. This study introduces the novel use of action integrals to examine stability, and the results provide testable predictions that round apples as compared to elongated or squat apples and larger elongated apples will need to travel farther before orientation is favored.

The wave annihilation technique and the design of nonreflective coatings

We develop theory and algorithms for the design of coatings which either eliminate or enhance reflection of waves from surfaces. For steady-state harmonic waves with continuous frequency spectrum that covers an arbitrarily prescribed frequency band, coatings are designed that essentially eliminate reflections of all frequencies within the band. Although we focus on acoustic waves in elastic media, the methods developed here can be adapted to electromagnetism or other phenomena governed by variants of the linear wave equation.To create a nonreflective coating which is to operate in a frequency band

The completeness of Maxwell's stress function representation

[Omega_0, Omega_1]

VARIATIONAL PROBLEMS IN WEIGHTED SOBOLEV SPACES ON NON-SMOOTH DOMAINS

, we select n frequencies, uniformly distributed in the band, and design an n-layer coating of given thickness that completely eliminates reflections of waves at these frequencies. We show that if n is large, then the reflectivity of the coating designed by our method is small for all frequencies in the band. More precisely, the reflectivity at an arbitrary frequency