When does the free boundary enter into corner points of the fixed boundary?

Our prime goal in this note is to lay the ground for studying free boundaries close to the corner points of a fixed, Lipschitz boundary. Our study is restricted to 2-space dimensions, and to the obstacle problem. Our main result states that the free boundary can not enter into a corner $x^0$ of the fixed boundary, if the (interior) angle is less than $\pi$, provided the boundary datum is zero close to the point $x^0$. For larger angles and other boundary datum the free boundary may enter into corners, as discussed in the text.