On the measure and the structure of the free boundary of the lower dimensional obstacle problem. (English) Zbl 1405.35257

Arch. Ration. Mech. Anal. 230, No. 1, 125-184 (2018); correction ibid. 230, No. 2, 783-784 (2018).
The authors deal with three properties of a solution to the following nonlinear thin obstacle problem. \[ \left\{\begin{aligned} &u(x',0)\geq 0\;\text{ for }(x',0)\in B'_R,\\ &u(x',x_{n+1})=u(x',-x_{n+1})\; \text{ for }x=(x',x_{n+1})\in B_R,\\ &\text{div}(|x_{n+1}|^a\nabla u(x))=0\;\text{ for }x\in B_R\setminus\{(x',0):u(x',0)=0\},\\ &\text{div}(|x_{n+1}|^a\nabla u(x))\leq 0\;\text{ in the sense of distribution in }B_R,\\ &u(x)=g(x)\;\text{ for }x\in\partial B_R, \end{aligned}\right.{(*)_a} \]
\(g\in\mathcal{A}_R=\{v\in H^1(B_R,|x_{n+1}|^a\mathcal{L}^{n+1}):v(x',x_{n+1})=v(x',-x_{n+1})\text{ and }v(x',0)\geq 0\}\) such that \(\mathcal{L}^{n+1}\) is the Lebesgue measure, \(B_R,B'_R\) are open balls of radius \(R\) in \(\mathbb R^{n+1}\) and \(\mathbb R^n\), respectively, \(\partial B_R\) represents the boundary of \(B_R\), and \(|a|<1\).
The particular case \((*)_0\) provides the Signorini problem in elasticity.
Let \(\Lambda(u)=\{(x',0)\in B'_R:u(x',0)=0\}\) be the coincidence set and we denote by \(\Gamma(u)\) the free boundary of \(\Lambda(u)\), i.e., the topological boundary of \(\Lambda(u)\). The first property states that the free boundary of a solution \(u\) of \((*)_a\) has locally finite \((n-1)\)-dimensional Minkowski content, i.e., for \(K\) a relative compact in \(B'_1\), the Lebesgue measure of \(\mathcal{T}_r(K\cap \Gamma(u))\) is bounded, up to a multiplicative constant relying on \(K\), by \(r^2\), where \(\mathcal{T}_r(E)=\text{dist}(\cdot,E)^{-1}(0,r)\subset \mathbb R^{n+1}\) such that \(E\) is a subset of \(\mathbb R^{n}\) (Theorem 1.1). The second property states that there exists \((M_i)_{i\in\mathbb N}\), a family comprised by at most countable \(C^1\)-regular \((n-1)\)-dimensional submanifolds in \(\mathbb R^{n+1}\), such that the \((n-1)\)-dimensional Hausdorff measure of the complement of \(\bigcup_{i\in\mathbb N} M_i\) in the free boundary of a solution of \((*)_a\) is zero (Theorem 1.2). The third property focuses on the blow-up and the frequency of a solution of \((*)_a\) at a point of the free boundary (Theorem 1.3). Thanks to a counter-example, the authors declared in [Arch. Ration. Mech. Anal. 230, No. 2, 783–784 (2018; Zbl 1466.35376)] that Proposition 8.2, regarding the classification of homogeneous solutions, is not correct.


35R35 Free boundary problems for PDEs
28A78 Hausdorff and packing measures
35R06 PDEs with measure


Zbl 1466.35376


Full Text: DOI arXiv


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