Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes

Raitums, Uldis. "Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 309-334. <http://eudml.org/doc/246050>.

@article{Raitums2002,
abstract = {We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems\begin\{gather\}\operatorname\{div\}\left(\sum \limits \_\{s=1\}^\{s\_0\}\sigma \_s(x)F\_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\text\{in\}\,\Omega ,\\ u=(u\_1,\dots , u\_m)\in H\_0^1(\Omega ;\{\bf R\}^m),\;\sigma =(\sigma \_1,\dots ,\sigma \_\{s\_0\})\in S, \end\{gather\}where $\Omega \subset \{\bf R\}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\lbrace \sigma \, measurable\,\mid \,\sigma _s(x)=0\;\mbox\{or\}\,1,\;s=1,\dots ,s_0,\;\sigma _1(x)+\cdots +\sigma _\{s_0\}(x)=1\rbrace $. We show that $WZ$ is the zero level set for an integral functional with the integrand $Q\mathcal \{F\}$ being the $\{\bf A\}$-quasiconvex envelope for a certain function $\mathcal \{F\}$ and the operator $\{\bf A\}=(\mbox\{curl,div\})^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda $ (defined by the operator $\{\bf A\}$) $Q\{\mathcal \{F\}\}$ coincides with the $\{\bf A\}$-polyconvex envelope of $\mathcal \{F\}$ and can be computed by means of rank-one laminates.},
author = {Raitums, Uldis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasilinear elliptic system; relaxation; A-quasiconvex envelope; -quasiconvex envelope; integral functional},
language = {eng},
pages = {309-334},
publisher = {EDP-Sciences},
title = {Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes},
url = {http://eudml.org/doc/246050},
volume = {7},
year = {2002},
}

TY - JOUR
AU - Raitums, Uldis
TI - Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 7
SP - 309
EP - 334
AB - We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems\begin{gather}\operatorname{div}\left(\sum \limits _{s=1}^{s_0}\sigma _s(x)F_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\text{in}\,\Omega ,\\ u=(u_1,\dots , u_m)\in H_0^1(\Omega ;{\bf R}^m),\;\sigma =(\sigma _1,\dots ,\sigma _{s_0})\in S, \end{gather}where $\Omega \subset {\bf R}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\lbrace \sigma \, measurable\,\mid \,\sigma _s(x)=0\;\mbox{or}\,1,\;s=1,\dots ,s_0,\;\sigma _1(x)+\cdots +\sigma _{s_0}(x)=1\rbrace $. We show that $WZ$ is the zero level set for an integral functional with the integrand $Q\mathcal {F}$ being the ${\bf A}$-quasiconvex envelope for a certain function $\mathcal {F}$ and the operator ${\bf A}=(\mbox{curl,div})^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda $ (defined by the operator ${\bf A}$) $Q{\mathcal {F}}$ coincides with the ${\bf A}$-polyconvex envelope of $\mathcal {F}$ and can be computed by means of rank-one laminates.
LA - eng
KW - quasilinear elliptic system; relaxation; A-quasiconvex envelope; -quasiconvex envelope; integral functional
UR - http://eudml.org/doc/246050
ER -