Layer potentials on rough boundaries and fractals
Oqtay Q. Azizov, Yevgeniy Guseynov · AIP conference proceedings · 2017
We are studying the layer potentials on rough boundaries in Rd, d>1, for Hölder classes. The new surface integral introduced by the second author allows extend the layer potentials properties to highly irregular boundaries like non-rectifiable Jordan curves, fractals, sets of finite perimeter boundaries and flat chains. Such surfaces could have the normal to the boundary nowhere, and we define the single and double layer potentials for continuous differential forms over a boundary of open bounded set that does not require a notion of boundary smoothness. Meanwhile, we show that for piecewise smooth boundaries the introduced layer potentials equivalent the classical single and double layer potentials. For domains that satisfy the condition which was introduced to describe the boundary integral existence for Hölder classes, we prove the Green’s identity and the Hölder continuity of the single and double layer potentials in the closure of domains.