Naslov
On some questions related to Memory effects in homogenisation and Propagation of singularities

Autori
Antonić, Nenad

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Department of Mathematics, College of Science

Mjesto
Pittsburgh

Datum
30.04

Godina
1992

Stranica
Viii+77

Mentor
Tartar, Luc

Ključne riječi
homogenisation; nonlocal effects; H-measures; symmetric hyperbolic systems; Maxwell's system

Sažetak
Two different problems arising in homogenisation theory have been addressed in the thesis: memory effects and propagation of oscillations. The theory of homogenisation studies the question: {; ; \sl If the solutions $u^\varepsilon$ of the problems ${; ; \cal A}; ; ^\varepsilon u^\varepsilon=f$ converge (weakly) to the function $u^0$, can an operator ${; ; \cal A}; ; ^0$ be found such that $u^0$ is a solution of the problem ${; ; \cal A}; ; ^0 u^0=f$, and is ${; ; \cal A}; ; ^0$ of the same type as ${; ; \cal A}; ; ^\varepsilon$? }; ; We study an example where the answer is negative. We take ${; ; \cal A}; ; ^\varepsilon := -a^\varepsilon(t) \partial_x^2 + b^\varepsilon(t) \partial_x + c^\varepsilon(t)$ and show that ${; ; \cal A}; ; ^0 := -a_{; ; \rm eff}; ; (t) \partial_x^2 + b_{; ; \rm eff}; ; (t) \partial_x + c_{; ; \rm eff}; ; (t)+ K \ast $ \ is an integrodifferential operator. The expression for $K$ is deduced under two different sets of assumptions --- bounds in L$^1$ or L$^2$. The L$^2$ setting uses Fourier transform and natural assumptions on the coefficients $a^\varepsilon, b^\varepsilon$ and $c^\varepsilon$ --- boundedness in L$^\infty$ and uniform ellipticity. In the L$^1$ setting additional assumptions are made. 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. A related problem appears in electromagnetism. In the homogenisation of the full system of Maxwell's equations, the above question can have either positive or negative answer. If the electrical conductivity is zero, the homogenised problem remains a hyperbolic system of differential equations. On the other hand, if we include the nonzero electrical conductivity, the answer is negative. In the periodic case (Sanchez-Palencia) it is possible to obtain the explicit formula for the kernel of the integral term. In the nonperiodic case, even though some partial results have been obtained, the complete answer is not known. The second problem considered in the thesis is an application of H-measures, a tool recently developed by Tartar. H-measures were introduced for homogenisation theory, but there are some promising applications in other fields as well. The H-measure is a Radon measure on the spherical bundle $\Omega\times S^{; ; n-1}; ; $ over the domain $\Omega\subseteq\Rn$ in consideration. We study the propagation properties of the H-measures associated to the solutions of symmetric systems, i.e.~the following system of partial differential equations for a vector function $\vu : \Omega \dstr \Rp $ ($\Omega \subseteq \R^{; ; n+1}; ; $, and $\mA^k$ are hermitian matrices): $ \mA^k\partial_k \vu + \mB\vu = \vf $. The symbol of the differential operator is $\mP(\mx, \mxi) = \xi_k\mA^k(\mx)$. In order to use H-measures, a sequence of problems should be introduced---replacing $\vu$ and $\vf$ in (1), for each $\eps>0$ by $\vu^\eps$ and $\vf^\eps$ respectively. If $\vu^\eps\dscon\vnul$ and $\vf^\eps\dscon\vnul$ (weakly in L$^2$), they define a $2\times2$ block matrix measure $\mmu$, $\mmu = \left[ \matrix{; ; \mmu_{; ; 11}; ; & \mmu_{; ; 12}; ; \cr \mmu_{; ; 21}; ; & \mmu_{; ; 22}; ; \cr}; ; \right]$, with blocks being $p\times p$ matrix measures. The localisation property reads: $\mP \mmu_{; ; 11}; ; = \mnul$ ($\mmu_{; ; 11}; ; $ is the H-measure defined by the sequence $(\vu^\eps)$ only). For the propagation property, the equation should be acted at by a scalar pseudodifferential operator $\cal A$ of order one, with a sufficiently smooth symbol $a$ ($a$ is a homogeneous function of the dual variable $\mxi = \nred{; ; \xi_}; ; \in \Rn\setminus \{; ; \mo\}; ; $ ). Defining $\psi(\mx, \mxi) := a(\mxi) w(\mx)$, where $w\in{; ; \rm C}; ; ^1_{; ; \rm c}; ; (\Omega)$ is a test function on the {; ; \sl physical space}; ; (the symbol $a$ is used as a test function on the {; ; \sl phase space}; ; \/), the propagation property reads: $$ \langle \mmu_{; ; 11}; ; , \{; ; \mP , \psi \}; ; + \psi \partial_k \mA^k -2\psi\mS\rangle + \langle 2 \Re \tr \mmu_{; ; 12}; ; , \psi \rangle = 0 \ ; , $$ where $\{; ; \mP , \psi \}; ; = \partial^l \mP\, \partial_l \psi - \partial^l \!\psi\, \partial_l \mP$ is the Poisson bracket, while $\mS$ is the hermitian part of $\mB$ ($2\mS=\mB+\mB^\ast$). The general result obtained is then applied to different systems, associated to the wave equation, Maxwell's equations and some equations that change type: Tricomi's equation and variants. In the latter case, the H-measure is not supported in the elliptic region ; it moves along the characteristics in the hyperbolic region, and bounces off the parabolic boundary, which separates the hyperbolic region from the elliptic region.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

Napomena
Http://www.genealogy.ams.org/id.php?id=85776

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