Preface Free boundary problems (FBPs) are considered today as one of the most important directions in the mainstream of the analysis of partial differential equations (PDEs), with an abundance of applications in various sciences and real world problems. In the past two decades, various new ideas, techniques, and methods have been developed, and new important, challenging problems in physics, industry, finance, biology, and other areas have arisen. The study of free boundaries is an extremely broad topic not only due to the diversity of applications but also because of the variety of the questions one may be interested in, ranging from modeling and numerics to the purely theoretical questions. This breadth presents challenges and opportunities! A particular direction in free boundary problems has been the study of the regularity properties of the solutions and those of the free boundaries. Such questions are usually considered very hard, as the free boundary is not known a priori (it is part of the problem!) so the classical techniques in elliptic/parabolic PDEs do not apply. In many cases the success is achieved by combining the ideas from PDEs with the ones from geometric measure theory, the calculus of variations, harmonic analysis, etc. Today there are several excellent books on free boundaries, treating var- ious issues and questions: e.g. [DL76], [KS80], [Cra84], [Rod87], [Fri88], [CS05]. These books are great assets for anyone who wants to learn FBPs and related techniques however, with the exception of [CS05], they date back two decades. We believe that there is an urge for a book where some of the most recent developments and new methods in the regularity of free boundaries can be introduced to the nonexperts and particularly to the graduate students starting their research in the field. This gap in the liter- ature has been partially filled by the aforementioned book of Caffarelli and ix
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