An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations

Authors

  • Rômulo Damasclin Chaves dos Santos Department of Physics, Technological Institute of Aeronautics, São Paulo, Brazil https://orcid.org/0000-0002-9482-1998
  • Jorge Henrique de Oliveira Sales State University of Santa Cruz – Department of Exact Sciences, Ilhéus, Bahia, Brazil https://orcid.org/0000-0003-1992-3748

DOI:

https://doi.org/10.18540/jcecvl10iss7pp18895

Keywords:

Irregular Domain, Integral Equations, Sobolev Spaces, Uniqueness and Regularity

Abstract

This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.

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Published

2024-07-01

How to Cite

Santos, R. D. C. dos, & Sales, J. H. de O. (2024). An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations. The Journal of Engineering and Exact Sciences, 10(7), 18895. https://doi.org/10.18540/jcecvl10iss7pp18895

Issue

Section

General Articles