Elliptic Partial Differential Equations From An Elementary Viewpoint: What The Laplacian Is Good For

This book offers a captivating journey through the world of elliptic partial differential equations (PDEs) and their surprising appearances across science, engineering, and even everyday life. Blending mathematical rigor with an accessible and engaging style, it introduces the Laplace operator and elliptic PDEs through diverse real-world phenomena, from physics and biology to artificial intelligence and image processing. Readers will explore fundamental topics such as diffusion, wave propagation, and fluid dynamics, alongside unexpected applications — including pattern formation in nature, traffic jams, quantum mechanics, and even the mathematics behind fighting pandemics and cancer. With a focus on both intuition and formalism, the book provides a wealth of examples, historical insights, and thought-provoking problems that make the subject come alive. Awarded the prestigious Book Prize of the Unione Matematica Italiana, this work has been praised as "a remarkable journey through delightful applications of elliptic PDEs, all while maintaining the highest standards of mathematical rigor." Whether you are a student, researcher, or simply curious about how mathematics shapes the world around us, this book will challenge and inspire you.

Serena Dipierro is Professor of Mathematics, Australian Research Council Future Fellow, and Fellow of the Australian Mathematical Society. She carried out her academic career in Santiago de Chile, Edinburgh, Magdeburg, Berlin, Melbourne, Milan, and Perth. She has been the recipient of the Australian Mathematical Society Medal, the Mahony–Neumann–Room Prize, the Bartolozzi Prize, the Christopher Heyde Medal, and the Book Prize of the Unione Matematica Italiana.

Enrico Valdinoci is Professor of Mathematics and Australian Laureate Fellow. He carried out his academic career in Pisa, Rome, Milan, Berlin, Melbourne, and Perth. He is a highly cited researcher, Ladyzhenskaya Lecturer, and has been awarded the James S W Wong Prize, the Mahony–Neumann–Room Prize, the Orazio Arena Prize, the Book Prize of the Unione Matematica Italiana, the Prize of the Bulletin of the Brazilian Mathematical Society, and the Amerio Gold Medal Prize.