Homepage of Bozhidar Velichkov

Maître de conférences, Université Joseph Fourier

I am working in the fields of Calculus of Variations and Partial Differental Equations. My recent research is concentrated mainly in the areas of:
Free Boundary Problems
Multiphase and Optimal Partition Problems
Spectral Optimization Problems

I am currently part of the group Calculus of Variations, Geometry and Images (CVGI) at Laboratoire Jean Kuntzmann (LJK).

Below you can find information on my
CV, List of Pubblications , Research topics, Research projects, Teaching,
as well as information on the CVGI seminar and some useful Links.


• In 2014 I was a post-doc at the Universtity of Pisa.

• From 2010 to 2013 I was a Ph.D. student at Scuola Normale Superiore and since 2012 also at Université de Savoie . My thesis was prepared under the direction of Giuseppe Buttazzo and Dorin Bucur.

• From 2005 to 2010 I was a student at SNS.


Free Boundary Problems

I am interested in the regularity of the solutions of free boundary and free discontinuity problems, which typically arise in the theory of Shape Optimization involving spectral functionals depending on the Dirichlet or Neumann Laplacian with perimeter or measure term. The analysis of the qualitative properties of the optimal sets is often an essential part of the existence theory, as one may see in the following paper.

Existence and Regularity for Spectral Optimization Problems with Perimeter Constraint

(with Guido De Philippis, Appl. Math. Optim. 69 (2) (2014), 199--231)

We prove the existence of an optimal set and the C1,α regularity of its boundary, for spectral functionals under a perimeter constraint, a model problem being

\min\Big\{\lambda_k(\Omega):\ Per(\Omega)=1,\ \Omega\subset\mathbb{R}^d,\ |\Omega|<\infty\Big\},

where λk is the k-th eigenvalue of the Dirichlet Laplacian on Ω. The key of our regularity result stands in the careful analysis of the sets of positive curvature in viscosity sense and an adaptation of the Bucur's Reduction Lemma. The combination of these techniques leads to the quasi-minimality, with respect to the perimeter, of the optimal sets, which then gives the regularity of the optimal domains.

In the paper

Lipschitz Regularity of the Eigenfunctions on Optimal Domains

(with Dorin Bucur, Dario Mazzoleni and Aldo Pratelli, Arch. Rat. Mech. Anal. 216 (1) (2015), 117--151)

we prove the long-standing open question concerning the Lipschitz continuity of the eigenfunctions on the domains, which are solutions of the problem

\min\Big\{\lambda_k(\Omega):\ \Omega\subset\mathbb{R}^d,\ |\Omega|=1\Big\},
where λk is the k-th eigenvalue eigenvalue of the Dirichlet Laplacian. We refine the techniques of Alt-Caffarelli and Briancon-Hayouni-Pierre and we prove that the eigenfunctions, which are quasi minimizers of the Dirichlet integral, are Lipschitz continuous. We then reason by approximation, using sequence of optimal sets, solutions of a specific obstacle problem, for which we can deduce the quasi-minimality of the corresponding eigenfunctions.

Our analysis proves, in particular, that there is a solution of the problem

\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega):\ \Omega\ open,\ \Omega\subset\mathbb{R}^d,\ |\Omega|=1\Big\},
and all the first k eigenfunctions are Lipschitz. We are currently working on the regularity of the free boundary of the optimal sets.

In the paper

A free boundary problem arising in PDE optimization

(with Giuseppe Buttazzo and Edouard Oudet, to appear in Calc.Var.PDE)

we consider a free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form
\displaystyle\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}\vert|\nabla u\vert|^2-fu\Big)\,dx.
We prove the existence of an optimal reinforcement and that it has some higher integrability properties. This problem is related to the classical elastic-plastic torsion problem. Some numerical simulations can be found here .


Multiphase and Optimal Partition Problems

The multiphase problems are a type of shape optimization problems, in which more domains are involved. The analysis of the boundary of the optimal sets requires some fine tools as the monotonicity formula of Alt-Caffarelli-Friedman and its variants and generalizations by Caffarelli-Jerison-Kenig, Conti-Terracini-Verzini and Bucur-Velichkov. In the paper

Multiphase Shape Optimization Problems

(with Dorin Bucur, SIAM J. Control Optim. 52(6) (2014), 3556--3591)

we study multiphase problems involving spectral functionals, the model problem being
\min\Big\{\sum_{i=1}^h\lambda_{k}(\Omega_i)+c|\Omega_i|:\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},

where λk is the k-th eigenvalue of the Dirichlet Laplacian and D is a given bounded open set. We prove a density estimate for the optimal sets and we establish the lack of three-phase boundary points through a multiphase version of the Caffarelli-Jerison-Kenig monotonicity formula. A big part of the paper is dedicated to the study of the emerging notion of energy subsolution, which appears to be an essential tool in the analysis of shape optimization and free boundary problems.

An essential tool in the study of multiphase problems is the monotonicity formula of Alt-Caffarelli-Friedman. In the paper

A Note on the Monotonicity Formula of Caffarelli-Jerison-Kenig

(Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat.Appl.)

we analyse the following Caffarelli-Jerison-Kenig version of the monotonicity formula
\prod_{i=1}^2\int_{B_r}\frac{|\nabla w_i|^2}{|x|^{d-2}}\,dx\le C\left(1+\int_{B_1}\frac{|\nabla w_1|^2}{|x|^{d-2}}+\int_{B_1}\frac{|\nabla w_2|^2}{|x|^{d-2}}\right)^2.

Our goal is to show that the above estimate holds also for non-continuous Sobolev functions wi whose distributional laplacian is bounded from below. We also give the detailed proof of the three-phase version of this formula, which appears to be a powerful tool in the study of optimal partition and multiphase problems.

In the paper

\min\Big\{\sum_{i=1}^h\lambda_{1}(\Omega_i)+c|\Omega_i|:\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},
where λ1 is the first eigenvalue of the Dirichlet Laplacian and D is a bounded open set in R2. We prove the Lipschitz regularity of the eigenfunctions and we study the free boundary of the optimal sets. Our analysis uses a refined version of the Alt-Caffarelli-Friedman monotonicity formula, which allows to exclude the presence of two-phase point on the boundary of D. We also provide some fine numerical results, based on a method introduced by Bourdin-Bucur-Oudet, confirming numerically the honeycomb conjecture of Caffarelli-Lin.


Spectral Optimization Problems

The optimization problems concerning the spectrum of the Laplacian are part of the classical problems in the analysis and are currently receiving a lot of attention. We survey some of the recent advances in this field in the paper


Some New Problems in Spectral Optimization

(with Giuseppe Buttazzo, Banach Center Publications)

The techniques of proving existence are quite involved and some fine variational tools are needed, as the Γ-convergence, introduced by De Giorgi, a concentration-compactness principle for operators and perturbation techniques in the spirit of Alt-Caffarelli.

Our first spectral optimization result has a fundamental role in the proof of the regularity of the state functions for the optimal sets for λk in Rd. In the paper

Spectral Optimization Problems with Internal Constraint

(with Dorin Bucur and Giuseppe Buttazzo, Ann.Inst.H.P., Vol.30 n.3 (2013), 477--495)

we consider the following shape optimization problem
\min\Big\{\lambda_k(\Omega)+|\Omega|:\ D\subset\Omega\subset\mathbb{R}^d\Big\},

where λk is the k-th eigenvalue of the Dirichlet Laplacian on Ω and the set D is a given obstacle to which we refer as internal constraint. We prove the existence of an optimal set using a concentration-compactness argument for Sobolev spaces combined with an Alt-Caffarelli-type perturbation technique. We show that the minima are bounded, have perimeter (for any k) and smooth boundary (for k=1). We also give a counterexample to the convexity of the optimal set.

We turn our attention to optimization in more general settings in the paper

Shape Optimization Problems on Metric Measure Spaces

(with Giuseppe Buttazzo, J.Funct.Anal. Vol.264 n.1 (2013),1--33)

where we prove the non-linear version of the classical Buttazzo-Dal Maso Theorem on the existence of a solution to the problem
\min\Big\{F\big(\lambda_1(\Omega),\dots,\lambda_k(\Omega)\big):\ \Omega\subset X,\ m(\Omega)\le1\Big\},

where (X,m) is a metric measure space and the spectrum (λ1(Ω),...,λk(Ω),...) of the Dirichlet Laplacian is variationally defined through the Rayleigh quotient on the Cheeger-Sobolev Space on (X,m). Our results apply to a large variety of frameworks as infinite dimensional Hilbert spaces with Gaussian measures, Finsler manifolds and Carnot-Caratheodory spaces.

In the paper

Optimal Potentials for Schrodinger Operators

(with Giuseppe Buttazzo, Augusto Gerolin and Berardo Ruffini, Journal de l'École polytechnique — Mathématiques, 1 (2014), p. 71-100)

we study spectral optimization problems for the Scrodinger operator -Δ+V under an integrability constraint of the potential V. We are able to prove existence results for very-general spectral functionals when we restrict the operator to a bounded domain. In the entire space Rd the question is more involved. When we optimize with respect to an energy functional (or the principal eigenvalue), the optimization problem is related to the ground state solution of nonlinear PDE with a sublinear term. In this special case we reduce the optimization problem to a ground state-type minimization problem, which gives the existence and then we study the growth at infinity of the optimal potentials.

We continue the analysis of the spectral optimization problems concerning Schrodinger operators in the paper

Spectral Optimization Problems for Potentials and Measures

(with Dorin Bucur and Giuseppe Buttazzo, SIAM J. Math. Anal. 46 (4) (2014), 2956--2986)

we consider two types of optimization problems: the generalized Kohler-Jobin problem
\min\Big\{\lambda_k(\mu):\ \mu\ capacitary\ measure\ in\ \mathbb{R}^d,\ T(\mu)=1\Big\},
where λk(μ) and T(μ) are the k-th eigenvalue and the torsional rigidity of the operator -Δ+μ and the minimization of the kth eigenvalue of the operator -Δ+V for trapping potentials satisfying integrability constraint of the form

Our proof of existence of minima is based on the study of the fine properties of the solutions U of the PDE -Δ U+μ U=f and, in particular, on a comparison principle "at infinity", obtained by a De Giorgi regularity technique and an analysis of viscosity solutions.

In many shape optimization problems an essential role for the existence of an optimal set is played by a given ambient space, which contains all the state functions of the various "shapes". A situation in which the above hypothesis does not hold is considered in the paper

Shape Optimization Problems for Metric Graphs

(with Giuseppe Buttazzo and Berardo Ruffini, ESAIM: COCV 20 n.1 (2014), 1--22)

where we study optimization problems for one-dimensional rectifiable sets. We consider functionals defined through differential operators that act directly on the graph, a model problem being
\min\Big\{E(\Gamma)\ :\ \Gamma\in A,\ H^1(\Gamma)=l\ \Big\},

here H1 denotes the one-dimensional Hausdorff measure and A is an admissible class of one-dimensional sets connecting some prescribed set of points D=(D1,...,Dk) in Rd. The cost functional E(Γ) is the Dirichlet energy ofΓ defined through the space of Sobolev functions on Γ, vanishing in the points of D. In order to obtain existence we extend the problem to the family of abstract metric graphs, which can be immersed in Rd in a way to connect the points from D.


The project VARIFORM (JUNE 2015 - DEC 2016) aims to attack several existence and regularity issues arising in shape optimization with respect to the spectrum of the Laplacian with Dirichlet and Neumann boundary conditions. The project is financed by Université Grenoble Alpes as part of the support program AGIR.

The following activities were entirely or partially supported by VARIFORM:

(*) 28.06.2015-04.07.2015 : Participation to the workshop
Shape optimization and spectral geometry

(*) 10-17.07.2015 : Visit to Università degli Studi di Torino.


24-25.09.2015, Chambery
Worshop on shape optimization and calculus of variations

22-23.10.2015, Paris

18-22.01.2016, Levico Terme
XXVI Convegno Nazionale di Calcolo delle Variazioni


Introduction to Ordinary and Partial Differential Equations

MAT 116

MAT 115

CVGI seminar

Cycle of seminars dedicated to the calculus of variations and its applications.
(Jointly organized with
Charles Dapogny)


Calculus of Variations and Geometric Measure Theory

Analysis of PDEs

Project ANR GAOS

Project ANR Optiform