In this talk I will present recent developments of the obstacle type problems, with various applications ranging
from: Industry to Finance, local to nonlocal operators, and one to multi-phases.
The theory has evolved from a single equation
$$
\Delta u = \chi_{u > 0}, \qquad u \geq 0
$$
to embrace a more general (two-phase) form
$$
\Delta u = \lambda_+ \chi_{u>0} - \lambda_- \chi_{u0$.
The above problem changes drastically if one allows $\lambda_\pm$ to have the incorrect sign (that appears in composite membrane problem)!
In part of my talk I will focus on the simple {\it unstable} case
$$
\Delta u = - \chi_{u>0}
$$
and present very recent results (Andersson, Sh., Weiss) that classifies the set of singular points ($\{u=\nabla u =0\}$) for the above problem.
The techniques developed recently by our team also shows an unorthodox approach to such problems, as the classical technique fails.
At the end of my talk I will explain the technique in a heuristic way.