## 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.

### MSC:

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

### Keywords:

nonlinear thin obstacle problem; Hausdorff measure

Zbl 1466.35376

DLMF
Full Text:

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