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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.
List of contents
- 1. Introduction and Main Results. - 2. Existence of Solutions, Qualitative Properties and Examples. - 3. Lipschitz Continuity of the Minimizers. - 4. Non-degeneracy of the Local Minimizers. - 5. Measure and Dimension of the Free Boundary. - 6. Blow-Up Sequences and Blow-Up Limits. - 7. Improvement of Flatness. - 8. Regularity of the Flat Free Boundaries. - 9. The Weiss Monotonicity Formula and Its Consequences. - 10. Dimension of the Singular Set. - 11. Regularity of the Free Boundary for Measure Constrained Minimizers. - 12. An Epiperimetric Inequality Approach to the Regularity of the One-Phase Free Boundaries.
About the author
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.
