Fictitious Domain Method for a Point Contact Problem for a Nonhomogeneous Body with a Thin Rigid Inclusion

A nonlinear problem formulated in a non-Lipschitz domain describing a pointwise contact of a nonhomogeneous $$2D$$ body with a thin rigid inclusion is considered. The domain boundary has one single cusp. A non penetration condition subject to possible contact interaction is imposed at a vertex of the cusp. Using a fictitious domain method the existence of a solution is proved. An auxiliary problem is formulated in a wider domain. Moreover, it is shown that solutions of the auxiliary and initial problems coincides.