On the regularity of the free boundary in the \(p\)-Laplacian obstacle problem
The paper is focused on the obstacle problem \(\min\{-\Delta_pu,u-\varphi\}=0\) in \(\Omega\subset \mathbb{R}^n\) for the \(p\)-Laplacian operator \(\Delta_pu=\operatorname{div} (|\nabla u|^{p-2}\nabla u)\), \(1<p<\infty\), where \(\Omega\) is a bounded smooth domain, and \(\varphi\) and \(g\) are given smooth functions. The authors study the regularity of the free boundary near those free boundary points \(x_0\in\partial\{u>\varphi\}\) at which \(\nabla \varphi(x_0)=0\). For \(p\not=2\) and \(\varphi(x)=-|x|^2\) in \(\mathbb{R}^2\), they construct global \(2\)-homogeneous solutions for the problem above with free boundaries having a corner at the origin. Then, under a concavity assumption for \(\varphi\), they prove that the free boundary is countably \((n-1)\) rectifiable, and present a nondegeneracy property for \(u\) at all free boundary points.
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