Calculus of Variations and Geometric Measure Theory

B. Velichkov

Regularity of the one-phase free boundaries

created by velichkov on 10 Jul 2019
modified on 30 Jun 2023


Lecture Notes

Inserted: 10 jul 2019
Last Updated: 30 jun 2023

Journal: Lecture Notes of the Unione Matematica Italiana, Springer
Volume: 28
Year: 2023
Links: Link to the PDF (open access) on the page of Springer


These notes are an introduction to the regularity theory for free boundary problems, the focus being 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 we present here are very recent and were developed in the context of different free boundary problems. We also give detailed proofs to several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories.