The problem on vibrations of a rectangular membrane subjected to a mechanical impact at a single point (Q2703961)

The paper under review deals with the study of the solvability of a simple mixed problem of vibrations of a rectangular membrane with fixed sides under mechanical impact at one point. Being at rest at an initial moment, the membrane is subjected to a mechanical impact of a load with given velocity. The load, after contact with the membrane, remains stuck at the impact point. The second initial condition is discontinuous. Therefore solutions of the problem cannot be classical. A generalized solution is understood to mean a continuous function whose first and second derivatives are represented as series summable almost everywhere in the sense of Cesàro, (C,1) and (C,2) respectively, and which satisfies almost everywhere the membrane equation, everywhere the first initial condition, everywhere but the impact point the second initial condition, and the integral condition containing the given velocity of the above load. The existence of such a generalized solution is proved.