On the Applications of Lebesgue Spaces to the Study of Partial Differential Equations

Abstract

In this work, we study the existence and uniqueness of solutions for some partial differential equations in Lebesgue and Sobolev spaces. First we recall basic notions on L p (Ω) spaces also their main properties such as completeness, H¨older and Minkowski inequalities, moreover weak convergence and compactness results. Then, we present the Lax–Milgram method, which is based on a variational formulation in a Hilbert space. This approach ensures existence and uniqueness of solutions under continuity and coercivity assumptions on the bilinear form. Finally, we study an alternative approach based on compactness arguments, using fundamental results such as the Rellich–Kondrachov theorem and the Banach–Alaoglu theorem to extract convergent subsequences and pass to the limit. These two methods are complementary: Lax–Milgram is more direct and efficient in Hilbert spaces, while compactness methods are more suitable for nonlinear and more general problems.