TY  - GEN
ID  - heidok8498
KW  - Dünne Strukturenhomogenization 
KW  -  periodic measures 
KW  -  monotone operators 
KW  -  degenerate parabolic equation 
KW  -  thin structures
Y1  - 2008///
TI  - Homogenization of quasilinear elliptic-parabolic equations with respect to measures
AV  - public
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/8498/
N2  - We investigate the homogenization of quasilinear elliptic and degenerate elliptic-parabolic equations arising in nonlinear filtration and flow transport in saturated as well as unsaturated porous media. The main focus of the thesis is to study these equations on general multidimensional structures, which we characterize by a periodic positive measure $\mu$ on ${\mathbb R}^d$. Our approach contains the classical framework of homogenization on perforated domains and, more importantly, the investigation of networks of arbitrary, possibly nonconstant dimension. To the aim of deriving effective macroscopic equations for nonlinear problems posed on these structures, we prove a new compactness result for bounded sequences $\{u_\varepsilon\}$ in the varying Sobolev spaces $H^{1,p}(\Omega,d\mu_\varepsilon)$, where the measures $\mu_\varepsilon$ are the nontrivial $\varepsilon$-rescalings of $\mu$, namely $\mu_\varepsilon(B):=\varepsilon^d \mu(\varepsilon^{-1}B)$, and where $\varepsilon$ is the typical microscopic length scale parameter. The singular measure approach will also be justified by a fattening ansatz, where a measure $\mu^\delta$, absolutely continuous with respect to the Lebesgue measure on ${\mathbb R}^d$, models   a thin reinforced structure of thickness $\delta >0$. We study in detail the two limit processes $\varepsilon \rightarrow 0$ and $\delta \rightarrow 0$ and show, at least for a large class of quasilinear problems, that the limits commute if the support of the singular measure $\mu$, the weak limit of $\mu^\delta$ as $\delta \rightarrow 0$, is sufficiently connected. On the other hand, by constructing explicit nontrivial counterexamples we will show that the limits do in general not commute on nonconnected structures, such that the homogenized equation will depend on the order we let the two parameters $\varepsilon$ and $\delta$ tend to zero.
A1  - Heuser, Philip
ER  -