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We study the Signorini problem near a fixed boundary, where the solution is "clamped down" or "glued." We show that in general the solutions are at least $C^{1/2}$ regular and that this regularity is sharp. We prove that near the actual points of contact of the free boundary with the fixed one the blowup solutions must have homogeneity $\kappa\geq 3/2$, while at the non-contact points the homogeneity must take one of the values: $1/2, 3/2, \ldots, m-1/2, \ldots$., Comment: 16 pages, 2 figues

The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points., Comment: 11 pges

In this paper we study regularity properties of the free boundary problem from superconductivity close to a fixed boundary. If the origin is a free boundary point, then we show that the free boundary touches the fixed boundary tangentially., Comment: 13 pages, 3 figures

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $\Delta u = \lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}, \lambda_\pm>0$. We prove that the free boundary touches the fixed one in (uniformly) tangential fashion if the boundary data $f$ and its first and second derivatives vanish at the touch-point., Comment: 14 pages, 3 figures