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Transactions of the American Mathematical Society

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Free boundary regularity for surfaces minimizing $ {\rm Area}(S)+c\,{\rm Area}(\partial S)$


Author: Edith A. Cook
Journal: Trans. Amer. Math. Soc. 290 (1985), 503-526
MSC: Primary 49F22
DOI: https://doi.org/10.1090/S0002-9947-1985-0792809-6
MathSciNet review: 792809
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Abstract: In $ {{\mathbf{R}}^n}$, fix a hyperplane $ Z$ and $ a\;(k - 1)$-dimensional surface $ F$ lying to one side of $ Z$ with boundary in $ Z$. We prove the existence of $ S$ and $ B$ minimizing $ \operatorname{Area}(S) + c\operatorname{Area}(B)$ among all $ k$-dimensional $ S$ having boundary $ F \cup B$, where $ B$ is a free boundary constrained to lie in $ Z$. We prove that except possibly on a set of Hausdorff dimension $ k - 2$, $ S$ is locally a $ {C^{1,\alpha }}$ manifold with $ {C^{1,\alpha }}$ boundary $ B$ for $ 0 < \alpha < 1/2$. If $ k = n - 1$, $ {C^{1,\alpha }}$ is replaced by real analytic.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0792809-6
Keywords: Free boundary, variational problem, flat chains, varifold, elliptic PDE, complementing boundary condition
Article copyright: © Copyright 1985 American Mathematical Society

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