Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 309-334.

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 Ω𝐑 n is a bounded Lipschitz domain, F s are strictly convex smooth functions with quadratic growth and S = { σ measurable 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 F s 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.

DOI : https://doi.org/10.1051/cocv:2002014
Classification : 49J45
Mots clés : quasilinear elliptic system, relaxation, A-quasiconvex envelope
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     title = {Relaxation of quasilinear elliptic systems via {A-quasiconvex} envelopes},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {309--334},
     publisher = {EDP-Sciences},
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     doi = {10.1051/cocv:2002014},
     zbl = {1037.49011},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002014/}
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Raitums, Uldis. Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 309-334. doi : 10.1051/cocv:2002014. http://www.numdam.org/articles/10.1051/cocv:2002014/

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