Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes

Uldis Raitums

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 7, page 309-334
  • ISSN: 1292-8119

Abstract

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We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems div s = 1 s 0 σ s ( x ) F s ' ( u ( x ) + g ( x ) ) - f ( x ) = 0 in Ω , u = ( u 1 , , u m ) H 0 1 ( Ω ; 𝐑 m ) , σ = ( σ 1 , , σ s 0 ) S , where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and S = { σ m e a s u r a b l e σ s ( x ) = 0 or 1 , s = 1 , , s 0 , σ 1 ( x ) + + σ s 0 ( x ) = 1 } . We show that WZ is the zero level set for an integral functional with the integrand Q being the A-quasiconvex envelope for a certain function and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone Λ (defined by the operator A) Q coincides with the A-polyconvex envelope of and can be computed by means of rank-one laminates.

How to cite

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Raitums, Uldis. "Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 309-334. <http://eudml.org/doc/90625>.

@article{Raitums2010,
abstract = { We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$ \begin\{array\}\{c\}\mbox\{div\}\left(\sum\limits\_\{s=1\}^\{s\_0\}\sigma\_s(x)F\_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\mbox\{in\}\,\Omega, u=(u\_1,\dots, u\_m)\in H\_0^1(\Omega;\{\bf R\}^m),\;\sigma=(\sigma\_1,\dots,\sigma\_\{s\_0\})\in S,\end\{array\} $$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\\{\sigma\, measurable\,\mid\,\sigma_s(x)=0\;\mbox\{or\}\,1,\;s=1,\dots,s_0,\;\sigma_1(x)+\cdots +\sigma_\{s_0\}(x)=1\\}$. We show that WZ is the zero level set for an integral functional with the integrand $Q\cal F$ being the A-quasiconvex envelope for a certain function $\cal F$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone Λ (defined by the operator A) $Q\{\cal F\}$ coincides with the A-polyconvex envelope of $\cal F$ 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.; quasilinear elliptic system; -quasiconvex envelope; integral functional},
language = {eng},
month = {3},
pages = {309-334},
publisher = {EDP Sciences},
title = {Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes},
url = {http://eudml.org/doc/90625},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Raitums, Uldis
TI - Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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 $$ \begin{array}{c}\mbox{div}\left(\sum\limits_{s=1}^{s_0}\sigma_s(x)F_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\mbox{in}\,\Omega, u=(u_1,\dots, u_m)\in H_0^1(\Omega;{\bf R}^m),\;\sigma=(\sigma_1,\dots,\sigma_{s_0})\in S,\end{array} $$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma\, measurable\,\mid\,\sigma_s(x)=0\;\mbox{or}\,1,\;s=1,\dots,s_0,\;\sigma_1(x)+\cdots +\sigma_{s_0}(x)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Q\cal F$ being the A-quasiconvex envelope for a certain function $\cal F$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone Λ (defined by the operator A) $Q{\cal F}$ coincides with the A-polyconvex envelope of $\cal F$ and can be computed by means of rank-one laminates.
LA - eng
KW - Quasilinear elliptic system; relaxation; A-quasiconvex envelope.; quasilinear elliptic system; -quasiconvex envelope; integral functional
UR - http://eudml.org/doc/90625
ER -

References

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  8. U. Raitums, Properties of optimal control problems for elliptic equations, edited by W. Jäger et al., Partial Differential Equations Theory and Numerical Solutions. Boca Raton: Chapman & Hall/CRC, Res. Notes in Math. 406 (2000) 290-297.  
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