A two-phase problem with a lower-dimensional free boundary
Summary: For a bounded domain \(D\subset \mathbb R^n\), we study minimizers of the energy functional \[ \int_{D}|\nabla u|^2\,dx + \int_{D \cap (\mathbb R^{n-1} \times \{0\} )} \lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u < 0\}} \,d\mathcal{H}^{n-1}, \] without any restriction on the function \(u\). One of the main results states that the free boundaries \[ \Gamma^+ = \partial \{u(\cdot, 0) > 0\}\quad \text{and}\quad \Gamma^- = \partial \{u(\cdot, 0) < 0\} \] 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.
scientific article; zbMATH DE number 6126404