Uniqueness and regularity of solutions of integral equations in irregular domains using fractional Sobolev spaces

Abstract

This paper investigates the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. While traditional studies often assume smooth boundaries, this research extends these results to domains with irregular boundaries. By utilizing Sobolev spaces, particularly fractional Sobolev spaces, and the properties of the Slobodetskii norm, a robust theoretical framework is developed. The main theorem of this study demonstrates that, under suitable conditions, the integral equation under consideration has a unique solution that inherits regularity properties. In other words, even in domains with irregular boundaries, it is possible to guarantee the existence of unique solutions with certain smoothness properties. This is crucial for the application of mathematical methods to real-world problems. These results represent a significant advancement in the mathematical understanding of boundary value problems in non-smooth domains. This has important implications, as many practical problems in physics and engineering involve domains with complex and irregular boundaries where traditional techniques are not applicable. By extending the results to irregular domains, this work opens new possibilities for the application of mathematical methods in more realistic and complex situations. The mathematical tools developed, based on fractional Sobolev spaces and the Slobodetskii norm, offer an innovative and effective approach to addressing these challenges. In summary, this study not only broadens the scope of problems that can be solved using integral equations but also provides a solid theoretical foundation for future research and applications in various scientific and technological fields. The findings presented here promise to significantly influence the development of new techniques and solutions in multiple areas of knowledge.