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Homepage of Bozhidar Velichkov

Professore Ordinario (Full Professor) at
Dipartimento di Matematica
Università di Pisa

I am working in the fields of
Calculus of Variations and Partial Differental Equations.
I am interested in the so-called Regularity Theory, which is the study of the fine structure of the solutions to variational and PDE problems.
Most of my current research is dedicated to Free Boundary Problems.

Selected
results

Optimal partition problems


• Boundary regularity of the free interface in spectral optimal partition problems
(with Roberto Ognibene) - Summary/Cover letter

The one-phase Bernoulli problem


• A capillarity one-phase Bernoulli free boundary problem
(with Lorenzo Ferreri, Giorgio Tortone)


• Regularity for one-phase Bernoulli problems with discontinuous
weights and applications (with Lorenzo Ferreri)


• Uniqueness of the blow-up at isolated singularities
for the Alt-Caffarelli functional (with Max Engelstein, Luca Spolaor)

Two-phase Bernoulli problems


• A one-sided two phase Bernoulli free boundary problem
(with Lorenzo Ferreri)


• (Quasi-)conformal methods in two-dimensional free boundary problems
(with Guido De Philippis, Luca Spolaor)


• Regularity of the optimal sets for the second Dirichlet eigenvalue
(with Dario Mazzoleni, Baptiste Trey) - Summary/Cover letter


• Regularity of the free boundary for the two-phase Bernoulli problem
(with Guido De Philippis, Luca Spolaor)
     - Summary/Cover letter
     - Extended abstract - Workshop Calculus of Variations (Oberwolfach)
     - Slides of my talks in Pisa and Levico


• An epiperimetric inequality for the regularity of some free boundary
problems: the 2-dimensional case (with Luca Spolaor)

Free boundary systems


• On the dimension of the singular set in optimization problems with
measure constraint (with Dario Mazzoleni, Giorgio Tortone)


• Regularity of the optimal sets for a class of integral shape functionals
(with Giuseppe Buttazzo, Francesco Paolo Maiale, Dario Mazzoleni, Giorgio Tortone)


• The boundary Harnack principle on optimal domains
(with Francesco Paolo Maiale, Giorgio Tortone)


• Epsilon-regularity for the solutions of a free boundary system
(with Francesco Paolo Maiale, Giorgio Tortone)


• Regularity of the free boundary for the vectorial Bernoulli problem
(with Dario Mazzoleni, Susanna Terracini)


• Regularity of the optimal sets for some spectral functionals

(with Dario Mazzoleni, Susanna Terracini)

Obstacle problems


• Direct epiperimetric inequalities for the thin obstacle problem and applications
(with Maria Colombo and Luca Spolaor)


• A logarithmic epiperimetric inequality for the obstacle problem
(with Maria Colombo and Luca Spolaor)


• On the logarithmic epiperimetric inequality - slides of my talks in
Krakow, Aveiro, Oberwolfach, Trento, and Milano