# zbMATH — the first resource for mathematics

The structure of the free boundary for lower dimensional obstacle problems. (English) Zbl 1185.35339
Consider the obstacle problem consisting in looking for the minimizer $$u(x)$$ of the Dirichlet integral over the unit ball $$B_1$$ in $$\mathbb{R}^n {(n\geq3)}$$ among the elements of the closed convex set $K=\{v\in H^1(B_1), v=\phi\,\, {\text{on}}\quad \partial B_1, v| _{x_n=0}\geq0\}$ where $$\phi$$ 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 $$\lambda(u)$$ is the subset of $$x_n=0$$ where $$u$$ vanishes and we are also interested in the free boundary $$F(u)$$, which is the boundary of the set $$\{u\geq\varphi\}\bigcap\{x_n=0\}$$. The paper is concerned with the study of $$F(u)$$, which is shown to be a $$C^{1, \alpha}(n-2)$$-dimensional surface in $$\mathbb{R}^{n-1}$$ near non-degenerate points.

##### MSC:
 35R35 Free boundary problems for PDEs 35J20 Variational methods for second-order elliptic equations
Full Text: