# Homepage of Bozhidar Velichkov, Ph.D.

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

I obtained Ph.D. degrees from Scuola Normale Superiore and Université de Grenoble with a thesis discussed on 8 November 2013 in Pisa and prepared at Scuola Normale Superiore and Université de Savoie
under the direction of Giuseppe Buttazzo and Dorin Bucur.
The manuscript was accepted for pubblication in Edizioni della Normale.

## Research topics

### 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.

#### (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

#### (with Dorin Bucur, Dario Mazzoleni and Aldo Pratelli)

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.

## Research topics

### 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

#### (with Dorin Bucur)

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

#### (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

#### Multiphase Optimization Problems for Eigenvalues: Qualitative Properties and Numerical Results (with Beni Bogosel)

we consider the multiphase problem
\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.

## Research topics

### 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

.

#### (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

#### (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

#### (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

#### (with Giuseppe Buttazzo, Augusto Gerolin and Berardo Ruffini)

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

#### (with Dorin Bucur and Giuseppe Buttazzo)

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
\int_{\mathbb{R}^d}V^{-\alpha}\,dx=1.

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

#### (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.

## CVAG

#### Seminari di Calcolo delle Variazioni e Analisi Geometrica Dipartimento di Matematica, Università di Pisa. (Jointly organized with Agnese Di Castro)

The program for 2013 and 2014.

The abstracts are also available on the CVGMT homepage.

You can contact the organizers at