## Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes.(English)Zbl 1037.49011

This paper deals with the weak closure $$WZ$$ for the set of solutions of a family of quasilinear elliptic systems in a bounded Lipschitz domain in $$\mathbb R^n$$. The main feature of the problem studied in the paper is the presence of a nonlinearity whose derivatives are strictly convex smooth functions with quadratic growth. The main result asserts that $$WZ$$ is the zero level set for an integral functional with the integrand $${\mathcal Q}\,{\mathcal F}$$ being the A-quasiconvex envelope for a certain function $${\mathcal F}$$ and the operator A=(curl,div)$$^m$$. Next, is is shown that if the nonlinearity has certain isotropic properties, then the characteristic cone $$\Lambda$$ (defined by the operator A) $${\mathcal Q}\,{\mathcal F}$$ coincides with the A-polyconvex envelope of $${\mathcal F}$$ and it can be computed by means of rank-one laminates.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 49M20 Numerical methods of relaxation type
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### References:

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