Variational inequalities for the fractional Laplacian
The interesting paper under review deals with the free boundary problem for the fractional Laplacian \[ \begin{cases} u\geq \psi & \text{in }\;\Omega,\\ (-\Delta)^su\geq f & \text{in}\;\Omega,\\ (-\Delta)^su= f & \text{in }\;\{u>\psi\},\\ u=0 & \text{in }\;\mathbb{R}^n\setminus\overline{\Omega}, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with \(n\geq 1,\) and \(s\in(0,1),\) the measurable function \(f\) and the distribution \(f\) are given. The authors show that, under quite mild assumptions on the data, the free boundary problem possesses a solution. The approach adopted relies on the study of broader class of problems regarding variational inequalities for the fractional Laplacian that admit solutions characterized as minimizers of suitable functionals.
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