Free boundary problems naturally arise in ice melting and iron behaviour under abrupt temperature changes during forging (quenching). These problems can be formulated as seeking a pair \( (\Omega, u) \), where \( \Omega \) represents the domain and \( u \) is a function satisfying a specific partial differential equation (PDE) in \( \Omega \).

Understanding the domain's geometry is crucial yet challenging, with open questions dating back to the 1980s. This lecture will cover some classical results in linear and nonlinear scenarios. Key differences between these scenarios will be discussed, highlighting the challenges of the nonlinear case and comparing the methods in each case. This comparison unveils some geometric results in the nonlinear case.