Hypersurfaces with mean Curvature given by an Ambient Sobolev Function

Abstract

We consider n-hypersurfaces Σ_j with interior E_j whose mean curvature are given by the trace of an ambient Sobolev function u_j ∊ W^1,p(ℝⁿ⁺¹)

(0.1) \bar H_Σj = u_jν_Ej on Σ_j,

where ν_Ej denotes the inner normal of Σ_j. We investigate (0.1) when Σ_j → Σ weakly as varifolds and prove that Σ is an integral n-varifold with bounded first variation which still satisfies (0.1) for u_j → u, E_j → E. p has to satisfy

p > 1/2 (n + 1)

and p ≥ 4/3 if n = 1. The difficulty is that in the limit several layers can meet at Σ which creates cancellations of the mean curvature.