Geodesic
Boundaries of Graphs of Harmonic Functions
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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Harmonic functions $u\colon\mathbb R^n\to\mathbb R^m$ are equivalent to integral manifolds of an exterior differential system with independence condition $(M,\mathcal T,\omega)$. To this system one associates the space of conservation laws $\mathcal C$. They provide necessary conditions for $g\colon\mathbb S^{n-1}\to M$ to be the boundary of an integral submanifold. We show that in a local sense these conditions are also sufficient to guarantee the existence of an integral manifold with boundary $g(\mathbb S^{n-1})$. The proof uses standard linear elliptic theory to produce an integral manifold $G\colon D^n\to M$ and the completeness of the space of conservation laws to show that this candidate has $g(\mathbb S^{n-1})$ as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in $\mathbb C^m$ in the local case.
Keywords: exterior differential systems; integrable systems; conservation laws; moment conditions. 
@article{SIGMA_2009_5_a67,
     author = {D. Fox},
     title = {Boundaries of {Graphs} of {Harmonic} {Functions}},
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     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a67/}
}
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D. Fox. Boundaries of Graphs of Harmonic Functions. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a67/

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