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In this work we consider an inhomogeneous two-phase obstacle-type problemdriven by the fractional Laplacian. In particular, making use of theCaffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulasand blow-up analysis, we provide a classification of the possible vanishingorders, which implies the strong unique continuation property. Moreover, weprove a stratification result for the nodal set, together with estimates on itsHausdorff dimensions, for both the regular and the singular part. The maintools come from geometric measure theory and amount to Whitney's ExtensionTheorem and Federer's Reduction Principle.

On a weighted two-phase boundary obstacle problem