Nenad Antonić

H-measures applied to symmetric systems *

H-measures were recently introduced by Tartar [Thmo] as a tool that might provide much better understanding of propagating oscillations. Partial differential equations of mathematical physics can (almost always) be written in the form of a symmetric system: where A* and B are matrix functions, while u is a vector unknown function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems (theorem 3). This result, combined with the localisation property ([Thmo]) is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation and Maxwells system. Particular attention is paid to the equations that change type: Tricomis equation and variants. The H-measure is not supported in the elliptic region; it moves along the characteristics in the hyperbolic region, and bounces of the parabolic boundary, which separates the hyperbolic region from the elliptic region. This work is supported in part by National Science Foundation grant 8803317 and by Army Research Office contract DAAL03-91-C-0023.

Memory effects in homogenisation: Linear second-order equations

AbstractThe theory of homogenisation treats the question: If the solutions uɛof the problemsAɛuɛ=f converge weakly to the function u0, can an operatorA0be found such that u0is a solution of the problemA0u0=f, and isA0of the same type asAɛ?We study an example where the answer is negative. We take Aɛ

Sequential laminates in multiple state optimal design problems

: = - a^\varepsilon (t){\text{ }}\partial _x^2 + b^\varepsilon (t){\text{ }}\partial _x + c^\varepsilon (t)

Connecting Classical and Abstract Theory of Friedrichs Systems via Trace Operator

and show that A0

On Certain Properties of Spaces of Locally Sobolev Functions

: = - a_{{\text{eff}}} (t){\text{ }}\partial _x^2 + b_{{\text{eff}}} (t){\text{ }}\partial _x + c_{{\text{eff}}} (t) + K *

On Some Properties of Homogenised Coefficients for Stationary Diffusion Problem

is an integrodifferential operator.The expression for K is deduced under two different sets of assumptions — bounds in L1 or L2. The L2 setting uses the Fourier transform and natural assumptions on the coefficients aɛ, bɛ and cɛ — boundedness in L∞ and uniform ellipticity. The answer in the L1 setting is obtained only under additional assumptions, which seem to be unnecessary.Finally, the description of the memory term is given for a problem on a bounded interval, by using the eigenfunction expansion and a representation theorem for Nevanlinna functions. In one space dimension the equation