Um estudo de funções harmônicas e princípio do máximo

Abstract

The aim of this monograph is to present the main properties of harmonic functions, making it easier for readers who are studying this subject for the first time. To achieve this, we have created a self-contained text that discusses the subject gradually. Initially, we will explore the relationship between harmonic functions defined in an open ball and the mean value property of these functions. Using this foundation, we will prove the maximum principle for harmonic functions and demonstrate that each harmonic function is infinitely differentiable (C 8). Subsequently, we will present gradient estimates for harmonic functions, which will lead to the conclusion that the only harmonic functions that can be bounded (from above or below) are constants. We will also prove the famous Harnack’s inequality. Another significant result we will prove is that if a continuous function is harmonic in the weak sense, then it is harmonic in the strong sense. Finally, we will introduce subharmonic functions and present a maximum principle for them. We will conclude the monograph by using the theory of subharmonic functions to derive some estimates for harmonic functions, which will be useful for studying the H¨older continuity of harmonic functions. It is worth mentioning that all the results discussed in this monograph can be found in [5]. Furthermore, to facilitate understanding, the monograph will begin with a brief review of basic analysis topics typically covered in undergraduate mathematics courses.