Estimativa de Carleson para o g-Laplaciano em domínios NTA

Abstract

This thesis investigates the Carleson estimate, a fundamental result in Harmonic Analysis, Potential Theory, and the Theory of Partial Differential Equations (PDEs), initially established by Lennart Carleson and proven essential for various fields. Generally, the Carleson estimate establishes an upper bound for positive PDE solutions that vanish on a portion of a domain’s boundary. This control is given by the function’s value at a fixed point, distant from the boundary, and a universal constant that depends only on the domain’s geometry and the PDE’s parameters. We discuss the profound relationship between this concept and the Boundary Harnack Inequality (or Boundary Harnack Principle), a result involving the quotient of two functions that, under the same hypotheses as the Carleson estimate, guarantees these functions decay to zero at the same rate as we approach the domain’s boundary. Research in the area has advanced significantly in generalizing these results to nonlinear operators. In this context, our work focuses on the g-Laplacian, an operator that includes a wide class of nonlinear elliptic PDEs. Building on recent developments that obtained the Carleson estimate for inhomogeneous solutions of the g-Laplacian in semi-ball type domains, the main objective of this thesis is to extend this estimate to NTA domains, a substantially more general geometric class, which involve considerable technical challenges. To this end, we will prove essential ingredients such as a refinement of the Harnack Inequality for uniform domains and a De Giorgi-type oscillation lemma up to the boundary. Additionally, this thesis presents the Boundary Harnack Inequality as an application of the obtained Carleson estimate, solidifying the relationship between the two results in our new context. Finally, we explore another important application: the establishment of an exponential growth estimate for solutions of the g-Laplacian in unbounded cylindrical domains with an NTA base, generalizing analogous growth theorems for other domains and operators in the literature.