Ill-posed problems in mechanics

Abstract

The notion of ill-posed initial and boundary value problems for partial differential equations was introduced by Hadamard, who also presented the first example of an ill-posed problem for a specific partial differential equation. At the same time, there are numerous examples of ill-posed problems in any field of mechanics.

Hadamard and some of his successors believed that any ill-posed problem has no physical meaning and such problems should not be posed.

The present paper contains several examples of ill-posed problems. We show that if one deals with an applied problem, then overcoming the ill-posedness mathematically can help one to improve the structure in practice, which justifies the study of ill-posed problems.