This open access book is an introduction to the regularity theory
for free boundary problems. The focus is on the one-phase Bernoulli
problem, which is of particular interest as it deeply influenced
the development of the modern free boundary regularity theory and
is still an object of intensive research. The exposition is
organized around four main theorems, which are dedicated to the
one-phase functional in its simplest form. Many of the methods and
the techniques presented here are very recent and were developed in
the context of different free boundary problems. We also give the
detailed proofs of several classical results, which are based on
some universal ideas and are recurrent in the free boundary, PDE
and the geometric regularity theories. This book is aimed at
graduate students and researches and is accessible to anyone with a
moderate level of knowledge of elliptical PDEs.
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