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We give an extension of the theory of relaxation of variational integrals inclassical Sobolev spaces to the setting of metric Sobolev spaces. Moreprecisely, we establish a general framework to deal with the problem of findingan integral representation for relaxed variational functionals of variationalintegrals of the calculus of variations in the setting of metric measurespaces. We prove integral representation theorems, both in the convex andnon-convex case, which extend and complete previous results in the setting ofeuclidean measure spaces to the setting of metric measure spaces. We also showthat these integral representation theorems can be applied in the setting ofCheeger-Keith's differentiable structure.

On the relaxation of variational integrals in metric Sobolev spaces