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Div-curl type theorem, \(H\)-convergence and Stokes formula in the Heisenberg group. (English) Zbl 1092.35008

The authors prove a div-curl type theorem in the Heisenberg group \(H^1\) and develop the corresponding \(H\)-convergence theory for second order elliptic differential operator in divergence form on \(H^1\). The main difficulty is in the definition of curl operator in an intrinsic way; the operator turns out to be a second order differential operator in the left invariant horizontal vector fields. As an evidence of the coherence of the proposed definition a Stokes type theorem is proved; moreover the relation with Rumin’s complex on contact manifold is considered.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
43A80 Analysis on other specific Lie groups
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
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