The Problem of Deformations of a Singular String with a Nonlinear Boundary Condition

We study by variational methods a two-dimensional model of deformations of a Stieltjes string having localized interactions with the environment. In this model, the deviation of any point of the string from the equilibrium position under the action of an external force is characterized by two coordinates. We assume that the ends of the string are elastically fixed. Moreover, the movement of one of string ends is limited by a bounded, closed, convex set \(C\) , lying in a plane perpendicular to the equilibrium position. Depending on the applied external force, this string end either remains an internal point of \(C\) or touches the boundary of \(C\) . This leads to a nonlinear boundary condition at the corresponding point. We establish the necessary and sufficient conditions for the extremum of the energy functional, prove the existence and uniqueness theorems of the solution; find an explicit formula for the solution and study its dependence on the size of the set \(C\) .