this book presents a comprehensive treatment of rearrangements and their applications in partial differential equations (pde). the author, bernhard kawohl, explores various types of rearrangements, including monotone decreasing, quasiconcave, symmetric decreasing, and starshaped rearrangements, as well as symmetrizations such as steiner and schwarz symmetrization. the book also covers circular and spherical symmetrization. in the second part, the focus shifts to maximum principles in pde, with discussions on the moving plane method, convexity of level sets, and concavity or convexity of functions. the content is structured into three main parts: introduction, rearrangements, and maximum principles. the book includes references, an index of examples and assumptions, and a subject index. the mathematics subject classification includes topics such as convexity, pde, and optimization. the book is intended for researchers and students in mathematics, particularly those working in pde and related areas. it provides a detailed analysis of the properties of rearrangements and their implications for the solutions of pde. the author's approach is rigorous and theoretical, with a focus on the mathematical foundations of rearrangements and their applications. the book is a valuable resource for understanding the role of symmetry and convexity in pde. it is well-organized and includes a variety of examples and assumptions to illustrate the concepts. the content is suitable for advanced undergraduate and graduate students in mathematics. the book is published by springer-verlag and is part of the lecture notes in mathematics series. the isbn numbers are provided for identification. the book is dedicated to the author's family.this book presents a comprehensive treatment of rearrangements and their applications in partial differential equations (pde). the author, bernhard kawohl, explores various types of rearrangements, including monotone decreasing, quasiconcave, symmetric decreasing, and starshaped rearrangements, as well as symmetrizations such as steiner and schwarz symmetrization. the book also covers circular and spherical symmetrization. in the second part, the focus shifts to maximum principles in pde, with discussions on the moving plane method, convexity of level sets, and concavity or convexity of functions. the content is structured into three main parts: introduction, rearrangements, and maximum principles. the book includes references, an index of examples and assumptions, and a subject index. the mathematics subject classification includes topics such as convexity, pde, and optimization. the book is intended for researchers and students in mathematics, particularly those working in pde and related areas. it provides a detailed analysis of the properties of rearrangements and their implications for the solutions of pde. the author's approach is rigorous and theoretical, with a focus on the mathematical foundations of rearrangements and their applications. the book is a valuable resource for understanding the role of symmetry and convexity in pde. it is well-organized and includes a variety of examples and assumptions to illustrate the concepts. the content is suitable for advanced undergraduate and graduate students in mathematics. the book is published by springer-verlag and is part of the lecture notes in mathematics series. the isbn numbers are provided for identification. the book is dedicated to the author's family.