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By adapting the Cheeger-Simons approach to differential cohomology, weestablish a notion of differential cohomology with compact support. We showthat it is functorial with respect to open embeddings and that it fits into anatural diagram of exact sequences which compare it to compactly supportedsingular cohomology and differential forms with compact support, in fullanalogy to ordinary differential cohomology. We prove an excision theorem fordifferential cohomology using a suitable relative version. Furthermore, we useour model to give an independent proof of Pontryagin duality for differentialcohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125(2003) 791]: On any oriented manifold, ordinary differential cohomology isisomorphic to the smooth Pontryagin dual of compactly supported differentialcohomology. For manifolds of finite-type, a similar result is obtainedinterchanging ordinary with compactly supported differential cohomology.

Cheeger-Simons differential characters with compact support and Pontryagin duality