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

Two physical applications of the Laplace operator perturbed on a set of zer o measure are suggested. The approach is based on the theory of self-adjoin t 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 th e model solution is an approximation of the corresponding "realistic" solut ion. The second application is the description of the time evolution of a o ne-dimensional quasi-Chaplygin medium, which can be reduced using a hodogra ph transform to the iii-posed problem of the Laplace operator perturbed on a set of codimension two in R-3. Stability and instability conditions are o btained.