CONICET | Buscador de Institutos y Recursos Humanos

We study the Two Membranes Problem for fully nonlinear operators both in the local (second order) and nonlocal setting. The problem arises when studying a ``bid an ask´´ model in mathematical finance where some asset has a price that varies randomly and a buyer and a seller have to agree on a price for a transaction to take place. The local/nolocal character of the problem, as well as the form of the operators considered, come precisely from the nature of the process. We give a mathematical formulation for the problem and prove existence of solutions in the viscosity sense via a penalization method. In the second order case we show the optimal $C^{1,1}$ regularity of solutions and provide an example showing that no regularity of the free boundary is expected to hold in general. In the nonlocal case we get $C^{2s}$ regularity for solutions ($2s-arepsilon$ for any $arepsilon>0$ if $s=1/2$). In order to achieve that, we prove regularity estimates for fully nonlinear nonlocal equations with bounded right hand side, a result that has interest on its own and is obtained combining a blow up argument with an appropriate Liouville-type theorem.