Overview

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.

Full Product Details

9783031132377

Table of Contents

Reviews

“The author presents the material in a way that is simple to understand. The proofs are self-contained, and the author also has an appendix to state the more elementary results in PDE/analysis/geometric measure theory. The recurring technology is Lebesgue’s differentiation theorem, and the book succeeds in presenting technical proofs in manageable steps.” (Emanuel Indrei, zbMATH 1558.35007, 2025)

“The structure of the book is developed around four central theorems ... .  In addition to detailed and rigorous proofs of these classical results, the author also presents some recent advances and methods in the context of related free boundary and variational problems. The text is self-contained and suitable as an introduction for graduate students with a strong background in elliptic PDEs, variational principles, and geometric analysis ... .” (Michael Eden, Mathematical Reviews, November 3, 2025) “The author presents the material in a way that is simple to understand. The proofs are self-contained, and the author also has an appendix to state the more elementary results in PDE/analysis/geometric measure theory. The recurring technology is Lebesgue’s differentiation theorem, and the book succeeds in presenting technical proofs in manageable steps.” (Emanuel Indrei, zbMATH 1558.35007, 2025)

Author Information

Bozhidar Velichkov is working in the fields of Calculus of Variations and Partial Differential Equations, in particular, his research is focused on the regularity and the local structure of the solutions to free boundary problems. He has several important contributions to the theory of the vectorial free boundary problems and developed new tools as the epiperimetric and the log-epiperimetric inequalities for free boundary problems.

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