Equações semilineares com não linearidade de absorção e medida como dado

Abstract

This work aimed to investigate the nonlinear Dirichlet problem with measure data and the difficulties that arise by reducing the regularity of the data. An analytical framework was established to study the existence, uniqueness, and regularity of solutions when the boundary data is a measure, generalizing classical formulations that require data in Lebesgue spaces. The approach is based on Sobolev space theory, the notion of capacity, and measure theory, employing tools such as the Newtonian potential, Kato's inequality, and the weak and inverse maximum principles. The text develops the necessary theoretical foundations to address the subject, among them: measure and integration theory (including the Radon-Nikodym-Lebesgue decomposition for \(\sigma\)-finite measures), properties of harmonic functions, Sobolev space theory (embeddings, trace, Poincar\'e inequalities), and the formulation of the Poisson equation with measure data in the distributional sense. The main results obtained include: the characterization of solutions as the sum of a Newtonian potential and a harmonic function; the equivalence between distributional and dual of the set of smooth functions vanishing at the domain boundary formulations of the Dirichlet problem; and the derivation of more general versions of the maximum principle (weak and inverse) in the context of measures, using the decomposition of a measure into diffuse and concentrated parts with respect to Sobolev capacity. These results consolidate the basis for the subsequent study of nonlinear problems.