Analysis of the Hilbertian Properties of the Sobolev Space \(H^1(\Omega)\)
Grace Nkwese Mazoni
Department of Mathematics and Computer Science, Faculty of Science and Technology, National Pedagogical University, Kinshasa, DRC.
Clara Paluku Kasoki
Department of Mathematics and Computer Science, Faculty of Science and Technology, National Pedagogical University, Kinshasa, DRC.
Emilien Loranu Londjiringa
Department of Mathematics Physics, Exact Sciences Section, ISP BUNIA, Ituri, DRC.
Camile Likotelo Binene *
Department of Mathematics and Computer Science, Faculty of Science and Technology, National Pedagogical University, Kinshasa, DRC.
*Author to whom correspondence should be addressed.
Abstract
This paper thoroughly examines the Hilbertian properties of the Sobolev space H1(\(\Omega\)). Based on the Lebesgue space L2(\(\Omega\)), H1(\(\Omega\)) is of particular importance in the analysis of functions with weak derivatives. The focus is on the Hilbertian structure of this space, which allows for the definition of a specific inner product and a rigorous verification of the associated completeness. These fundamental characteristics facilitate a detailed study of convergence, continuity, and orthogonality of functions in H1(\(\Omega\)), thereby enhancing its relevance for solving various complex mathematical problems.
This paper aims to address gaps identified in the existing literature, where the explicit verification of these essential properties is sometimes omitted, thus compromising mathematical rigor and the applicability of results in various contexts.
Keywords: Hilbert space, sobolev space \(H^1(\Omega)\), lebesgue space \(L^2(\Omega)\), distribution space \(D'(\Omega)\), inner product, completeness, convergence, literature review