Harmonic functions on metric measure spaces are realized as weak limits of graph minimizers and shown to minimize a nonlinear energy on the Newton-Sobolev space that arises as the Gamma-limit of the discrete graph energies.
Maz’ya,Sobolev spaces with applications to elliptic partial differential equations.Second, revised and augmented edition
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Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs
Harmonic functions on metric measure spaces are realized as weak limits of graph minimizers and shown to minimize a nonlinear energy on the Newton-Sobolev space that arises as the Gamma-limit of the discrete graph energies.