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Part I Weakly differentiable functions  Part II PDEs on varifolds  15 Sobolev spaces ... 75  16 Preliminaries ... 82  17 Maximum estimates ... 86  18 Second order flatness of Lebesgue spaces for integral varifolds withsubcritical integrability of the mean curvature ... 94  19 Second order differentiability of the support of integral varifolds withcritical integrability of the mean curvature ... 100  20 Harnack inequality ... 108  Part III The generalised Gauss map  21 Points of finite lower density ... 118  22 Points of infinite density ... 130  23 Area formula for the generalised Gauss map ... 137  24 A sharp lower bound on the mean curvature integral with critical power forintegral varifolds ... 138  Appendix  A Monotonicity identity ... 139  B An observation concerning the Lusin property ... 140

A sharp lower bound on the mean curvature integral with critical power for integral varifolds