Regularity of the One-phase Free Boundaries

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.

• - 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.