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Hypersurfaces with mean curvature given by an ambient Sobolev function. (English) Zbl 1055.49032

Summary: We consider \(n\)-hypersurfaces \(\Sigma_j\) with interior \(E_j\) whose mean curvature are given by the trace of an ambient Sobolev function \(u_j\in W^{1,p}(\mathbb{R}^{n+1})\) \[ \vec{\mathbf H}_{\Sigma_j} =u_j\nu_{E_j}\quad\text{on }\Sigma_j,\tag{0.1} \] where \(\nu_{E_j}\) denotes the inner normal of \(\Sigma_j\). We investigate (0.1) when \(\Sigma_j\to\Sigma\) weakly as varifolds and prove that \(\Sigma\) is an integral \(n\)-varifold with bounded first variation which still satisfies (0.1) for \(u_j\to u\), \(E_j\to E\). \(p\) has to satisfy \(p>\tfrac 12(n+1)\) and \(p\geq\frac 43\) if \(n=1\). The difficulty is that in the limit several layers can meet at \(\Sigma\) which creates cancellations of the mean curvature.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
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