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Firstly, this paper establishes useful forms of the remainder term ofHardy-type inequalities on general domains where the weights are functions ofthe distance to the boundary. For weakly mean convex domains we use theresulting identities to establish nonexistence of extremizers for and improveknown sharp Hardy inequalities. Secondly, we establish geometricallyinteresting remainders for the Davies-Hardy-Tidblom inequalities for the meandistance function, as well as generalize and improve several Hardy typeinequalities in the spirit of Brezis and Marcus and spectral estimates ofDavies. Lastly, we apply our results to obtain Sobolev inequalities fornon-regular Riemannian metrics on geometric exterior domains.

$L^p$-Hardy identities and inequalities with respect to the distance and mean distance to the boundary