The two membranes problem for fully nonlinear local and nonlocal operators




Vivas, Hernán Agustín

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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 [superscript 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 C²[superscript s] regularity for solutions (2s − ε for any ε > 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



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