Two physical applications of the Laplace operator perturbed on a null set

Two physical applications of the Laplace operator perturbed on a set of zero measure are suggested. The approach is based on the theory of self-adjoint extensions of symmetrical operators. The first application is a solvable model of scattering of a plane wave by a perturbed thin cylinder. “Nonlocal” extensions are described. The model parameters can be chosen such that the model solution is an approximation of the corresponding “realistic” solution. The second application is the description of the time evolution of a one-dimensional quasi-Chaplygin medium, which can be reduced using a hodograph transform to the ill-posed problem of the Laplace operator perturbed on a set of codimension two in $\mathbf R^3$. Stability and instability conditions are obtained.