Introdução às funções harmônicas

Abstract

Harmonic functions are primordial objects of study for modern mathematics, particularly in the area of analysis and EDP, but also in other areas of knowledge, such as physics. Aiming at the deepening of such a class of functions, equivalences are demonstrated in relation to the definition of such functions, thus introducing concepts such as "weakly harmonic function"and "mean-value property". In addition, results are obtained about the regularity of such functions, proving that they are in fact "soft"and, still more, "analytical". In the middle of the demonstrations we obtain estimates in relation to such functions and their derivatives, properties related to their maximum and minimum (maximum principle), as well as results such as "Harnack’s inequality". Then a special kind of harmonic function, the fundamental solution of Laplace’s equation, is analyzed. From it we obtain the representation formula using Green’s function, Green’s function and, finally, the solution of the Problem of Dirichlet for balls. In the middle of the demonstrations we use previous results such as the "Divergence Theorem", Dominated Convergence Theorem, Weierstrass Theorem, Clairaut-Schwarz Theorem, and Mean Value Theorem without, however, demonstrating them.