Abstract
We prove that integral n-varifolds μ in codimension 1 with \( H{\;_{\mu }}\; \in \;L_{{loc}}^p\left( \mu \right),\;p\; > \;n \), p > n, p ≥ 2, have quadratic tilt-excess decay tiltexμ(x, ϱ, T x μ) = O x (ϱ2) for μ-almost all x. This regularity estimate is used to establish a general convergence procedure for hypersurfaces Σ j with interior E j whose mean curvatures are given by the trace of ambient Sobolev functions \( \overrightarrow {{H_{{{\Sigma_j}}}}} = {u_j}{\nu_{{{E_j}}}} \) on Σ j , where ν Ej denotes the inner normal of Σ j .
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Dedicated to Olga A. Ladyzhenskaya on her birthday
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Schätzle, R. (2002). A Geometric Regularity Estimate via Fully Nonlinear Elliptic Equations. In: Birman, M.S., Hildebrandt, S., Solonnikov, V.A., Uraltseva, N.N. (eds) Nonlinear Problems in Mathematical Physics and Related Topics I. International Mathematical Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0777-2_19
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DOI: https://doi.org/10.1007/978-1-4615-0777-2_19
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