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Here’s some references for PDEs if you’re interested in learning more. First, some general PDE books, good if you’re new to the field.

  • Partial Differential Equations by L.C. Evans. Excellent introduction to PDEs, but also useful as a reference and a survey of PDEs.
  • Partial Differential Equations by J. Jost. I found this book after I already knew most of the stuff in it, which is a shame since its exposition is really great. In particular, its coverage of the Moser-Harnack Inequality and the De Giorgi-Nash-Moser theory is one of the most accessible I’ve seen.
  • Partial Differential Equations in Action by S. Salsa. Recent book with a slightly different focus than the previous two. Lots of examples and excellent explanations of how a PDE arises from an application.

Free boundary books are less common, both because the field is not as old and because the results are more technical and specialized. In the same spirit I’ll give more specialized recommendations.

Variational Inequalities and Free Boundary Problems by A. Friedman is a gigantic (seriously, it’s like 700+ pages) book on the subject. With the exception of the first couple of chapters, the book chapters are pretty self-contained. Which is good, since like I said, it’s huge. If you’re looking for a specific free boundary problem to start with, I highly recommend the ‘porous dam problem’ from this book (it is found in other works as well).

You can also check out An Introduction to Variational Inequalities and Their Applications by D. Kinderlehrer and G. Stampacchia. This is probably a better reference than the Friedman book if you’re not as familiar with Hilbert and Banach space theory, though it is not as comprehensive as the Friedman book.

Lastly, it you’re seriously interested in the subject here’s a couple of free boundary books you can check out:

First, Regularity of Free Boundaries in Obstacle-Type Problems by A. Petrosyan, H. Shahgholian, and N. Uraltseva is a good, accessible (this is a relative term at this point), and very recent work that does a great job of not only collecting results that were previously scattered in papers, but also presenting these results as a cohesive theory.

Last but by no means least we have A Geometric Approach to Free Boundary Problems by L. Caffarelli and S. Salsa. I’m not going to say this is an easy read, but it contains ideas that, to my knowledge, are not presented anywhere else save the original papers. Papers which, by the way, are exceedingly important in the field of free boundaries. It also has a really good ‘appendix’ for results about harmonic and caloric functions on Lipschitz domains.