For certain PDEs, it is not always possible to find smooth solutions; hence traditionally, we tend to relax the requirement of smoothness to finding weak (distributional) solutions to those PDEs, using the theory of Sobolev spaces. In dealing with Sobolev spaces, especially for unbounded regions, we encounter functions that are locally integrable but are not globally integrable, but have globally finite Sobolev energy. Such functions are called Dirichlet-Sobolev functions. Constant functions are certainly of this kind, with zero energy. It is natural to ask whether such functions are always, after subtracting a suitable constant that may depend on the function, globally integrable. The focus of this talk is to present results related to this question in the context of complete metric measure spaces (including Riemannian manifolds and Carnot-Caratheodory spaces) equipped with a locally doubling measure supporting a local Poincare type inequality.

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