*(English)*Zbl 1185.35339

Consider the obstacle problem consisting in looking for the minimizer $u\left(x\right)$ of the Dirichlet integral over the unit ball ${B}_{1}$ in ${\mathbb{R}}^{n}(n\ge 3)$ among the elements of the closed convex set

where $\varphi $ is a smooth function which is supposed to be positive on $\partial {B}_{1}$, intersected with ${x}_{n}=0$ and to assume also negative values. The coincidence set $\lambda \left(u\right)$ is the subset of ${x}_{n}=0$ where $u$ vanishes and we are also interested in the free boundary $F\left(u\right)$, which is the boundary of the set $\{u\ge \phi \}\bigcap \{{x}_{n}=0\}$. The paper is concerned with the study of $F\left(u\right)$, which is shown to be a ${C}^{1,\alpha}(n-2)$-dimensional surface in ${\mathbb{R}}^{n-1}$ near non-degenerate points.