ELLIPTIC AND PARABOLIC EQUATIONS WITH L 1 DATA

Abstract

This dissertation addresses the study of elliptic and parabolic partial differential equations with data in the space L 1 (Ω), where Ω ⊂ RN is a bounded domain. Unlike the classical setting, which typically assumes sufficiently smooth, the L 1 framework poses significant analytical challenges due to the absence of compactness and regularity. We investigate the existence, uniqueness, and regularity of solutions in the weak sense. The analysis requires a suitable functional framework for L 1 data, involving tools from the theory of Sobolev spaces, truncation techniques, and a priori estimates. These methods allow us to extend classical results and provide a rigorous foundation for studying PDEs under minimal regularity assumptions.