On representation formulas and radiation conditions (Q2785654)

The method of integral representations of solutions is widely known and used in the theory of elliptic boundary value problems. The article concentrates on such representations in exterior domains. The authors consider rather general elliptic operators having fundamental solutions. What is called ``radiation condition'' means a special behaviour at infinity guaranteeing the validity of the representation formulas in exterior domains. The results obtained are quite general. In particular, for fundamental solutions in \(\mathbb{R}^n\) the following is proved: NEWLINENEWLINENEWLINEEvery fundamental solution satisfies its own radiation condition. -- If a solution can be represented by any combination of boundary or volume potentials, then it can be represented by its own Cauchy data on a given surface. -- There is a one-to-one correspondence between radiation conditions and fundamental solutions for any given elliptic differential operator. NEWLINENEWLINENEWLINEThis is generalized also to the case of an exterior domain. Some other questions as well as a series of interesting examples are considered.