A two-phase problem with a lower-dimensional free boundary

Mark Allen; Arshak Petrosyan

doi:10.4171/ifb/283

JournalsifbVol. 14, No. 3pp. 307–342

A two-phase problem with a lower-dimensional free boundary

Mark Allen
Purdue University, West Lafayette, USA
Arshak Petrosyan
Purdue University, West Lafayette, USA

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Abstract

For a bounded domain $D \subset R^{n}$ , we study minimizers of the energy functional

J (u) = \int_{D} ∣\nabla u ∣^{2} d x + \int_{D \cap (R^{n - 1} \times {0})} λ^{+} χ_{{u > 0}} + λ^{-} χ_{{u < 0}} d H^{n - 1},

without any sign restriction on the function $u$ . One of the main result states that the free boundaries

Γ^{+} = \partial {u (\cdot, 0) > 0} and Γ^{-} = \partial {u (\cdot, 0) < 0}

never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension $n = 3$ the free boundaries are $C^{1}$ regular on a dense subset.

Cite this article

Mark Allen, Arshak Petrosyan, A two-phase problem with a lower-dimensional free boundary. Interfaces Free Bound. 14 (2012), no. 3, pp. 307–342

DOI 10.4171/IFB/283