Geodesic
The de Rham cohomology through Hilbert space methods
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 455-465

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We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler–Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.
Keywords: De Rham complex, cohomology, Hodge theory, Neumann problem. 
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Ihsane Malass; Nikolai Tarkhanov. The de Rham cohomology through Hilbert space methods. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 455-465. http://geodesic.mathdoc.fr/item/JSFU_2019_12_4_a7/