Introduction to partial differential equations

Abstract

The purpose of this book is to serve students, teachers, and researchers who seek a clear and accessible introduction to modern methods in Partial Differential Equations (PDEs), particularly in regions where advanced mathematical material is not always readily available. Over the past decades, I have taught these topics in Brazil using lecture notes that grew organically from classroom experience. These notes have now been reorganized, expanded, and translated into English with a single goal in mind: to other mathematical knowledge to readers in developing countries, especially in some poor countries all over the world, where access to updated material is often limited. The topics presented here range from the classical linear theory to nonlinear analysis, monotonicity methods, compactness techniques, evolution equations, and a large variety of applied PDE models. Every chapter was written with great care for clarity and pedagogy. Technical proofs are included when necessary, but always with an effort to preserve intuition and motivation. Many of the problems re ect real questions arising from physics, engineering, and contemporary mathematical research. This book is also the result of a personal commitment. I firmly believe that mathematics is a universal language - one that should not be restricted by geographical or economic barriers. The possibility of translating these materials into English and making them accessible to students around the world is, to me, an act of gratitude and service. If these pages help even a single student develop confidence and inspiration to pursue further studies, then this work will have fulfilled its purpose. I am deeply grateful to Valéria Neves Domingos Cavalcanti, whose support, friendship, and mathematical insight have shaped much of my academic life. My thanks also go to the countless students whose questions and enthusiasm gave life to these notes. It is my sincere hope that this book becomes a bridge - connecting people, ideas, and opportunities - and that it might illuminate the path for those who, like me, believe in the transformative power of education.