Melting of ice, meatloaf and wall problems: an introduction to free boundary problems

The Dirichlet problem for the Laplace operator is the most basic example of a boundary value problem: given a domain \$\Omega\subset \mathbb{R}^n\$, a boundary value \$\varphi\in C(\partial \Omega)\$ and \$f\in C(\Omega)\$, one seeks a function \$u\in C^2(\Omega)\cap C(\overline \Omega)\$ such that \$\Delta u = f\$ in \$\Omega\$, and \$u = \varphi\$ on \$\partial \Omega\$.

 

In many problems arising from the applied sciences the situation is more delicate: the domain \$\Omega\$ is not completely known a priori; only a portion \$\Sigma \subset \partial \Omega\$ of its boundary is prescribed. An example of such a situation consists of solving \$\Delta u = f\$ in \$\Omega\$ where \$u = \varphi\$ on \$\Sigma\$. On the a priori unknown remaining part of the boundary, \$\Gamma = \partial \Omega \setminus \Sigma\$, one imposes additional conditions. This is a free boundary problem; solving it means finding not only the function \$u\$ but also the free boundary \$\Gamma\$. Another example of a free boundary problem is the Stefan problem, which describes the temperature distribution in a medium undergoing a phase change, for example melting of ice.

 

Free boundary problems arise naturally in a number of physical phenomena and deal with solving partial differential equations (PDEs) in a domain \$\Omega\$, a part of whose boundary is unknown in advance. In this talk we will see examples of free boundary problems, questions people are interested in answering, and techniques used in the field to address those questions.

 

P.S. No walls will be built during this talk.