Pablo Raúl Stinga, Associate Professor of Mathematics, has recently published the graduate level textbook “Regularity Techniques for Elliptic PDEs and the Fractional Laplacian” (Chapman and Hall/CRC Press, New York, June 2024, 334pp). Stinga’s textbook presents the most important regularity techniques, both classical and modern, for elliptic PDEs and for nonlocal equations of fractional order. Its foreword was co-written by 2023 Abel Prize awardee Luis A. Caffarelli and harmonic analysis renowned expert José L. Torrea.

Partial differential equations (PDEs) are at the core of mathematical modeling. Scientific areas like engineering, biology, soil science, finance and many other use PDEs to understand real world-phenomena. Typically, solutions to these equations are easily shown to exist in some weaker formulation of the models. Then the problem arises as to whether or not the solution actually has stronger properties (that is, *regularity*) to guarantee that it satisfies the original equations in the classical sense. Showing that solutions are *regular* is a difficult and exciting task.

Stinga’s graduate textbook will be helpful for students and researchers to bridge the gap between upper undergraduate PDE texts (like those by Evans or Salsa) and advanced research monographs in the area (like the one by Caffarelli-Cabré). The book emphasizes ideas and the development of a geometric intuition. At the same time, the text is fully rigorous and self-contained.

The textbook was conceived out of sets of lecture notes Stinga wrote for a mini-course on the fractional Laplacian he gave in Argentina in 2014, a graduate course on elliptic regularity he taught at The University of Texas at Austin in 2015, and on successive graduate and reading courses for students. The textbook is suitable for a semester course and it will be adopted for PDE II (MATH 6560), Harmonic Analysis (MATH 6310) and topics courses at ISU.