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Bozhidar Velichkov

Publications and source records attributed to Bozhidar Velichkov.

70 records · Page 4Linked to original sources
arXiv2016

A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^hλ_1(Ω_i)+α|Ω_i|:\ Ω_i\ \hbox{open},\ Ω_i\subset D,\ Ω_i\capΩ_j=\emptyset\Big\},$$ where $α>0$ is a given constant and $ D\subset\Bbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide numerical results for the optimal partitions.

math.OC↗
arXiv2016

Worst-case shape optimization for the Dirichlet energy

We consider the optimization problem for a shape cost functional $F(Ω,f)$ which depends on a domain $Ω$ varying in a suitable admissible class and on a "right-hand side" $f$. More precisely, the cost functional $F$ is given by an integral which involves the solution $u$ of an elliptic PDE in $Ω$ with right-hand side $f$; the boundary conditions considered are of the Dirichlet type. When the function $f$ is only known up to some degree of uncertainty, our goal is to obtain the existence of an optimal shape in the worst possible situation. Some numerical simulations are provided, showing the difference in the optimal shape between the case when $f$ is perfectly known and the case when only the worst situation is optimized.

math.OC↗
arXiv2015

BV Estimates in Optimal Transportation and Applications

In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well. In particular, in the case f = 1 (projection onto a set of densities with an L^\infty bound) we precisely prove that the total variation of the projection does not exceed the total variation of the projected measure. This is an estimate which can be iterated, and is therefore very useful in some evolutionary PDEs (crowd motion,. . .). We also establish some properties of the Wasserstein projection which are interesting in their own, and allow for instance to prove uniqueness of such a projection in a very general framework.

math.OC↗
arXiv2015

A free boundary problem arising in PDE optimization

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 $$\sup_{\int_Dθ\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+θ}{2}|\nabla u|^2-fu\Big)\,dx.$$ We prove the existence of an optimal reinforcement $θ$ and that it has some higher integrability properties. We also provide some numerical computations for $θ$ and $u$.

math.OC↗
arXiv2014

The spectral drop problem

We consider spectral optimization problems of the form $$\min\Big\{λ_1(Ω;D):\ Ω\subset D,\ |Ω|=1\Big\},$$ where $D$ is a given subset of the Euclidean space $\mathbb{R}^d$. Here $λ_1(Ω;D)$ is the first eigenvalue of the Laplace operator $-Δ$ with Dirichlet conditions on $\partialΩ\cap D$ and Neumann or Robin conditions on $\partialΩ\cap\partial D$. The equivalent variational formulation $$λ_1(Ω;D)=\min\left\{\int_Ω|\nabla u|^2\,dx+k\int_{\partial D}u^2\,d\mathcal{H}^{d-1}\ :\ u\in H^1(D),\ u=0\hbox{ on }\partialΩ\cap D,\ \|u\|_{L^2(Ω)}=1\right\}$$ reminds the classical drop problems, where the first eigenvalue replaces the total variation functional. We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.

math.AP↗
arXiv2013

Shape optimization problems on metric measure spaces

We consider shape optimization problems of the form $$\min\big\{J(Ω)\ :\ Ω\subset X,\ m(Ω)\le c\big\},$$ where $X$ is a metric measure space and $J$ is a suitable shape functional. We adapt the notions of $γ$-convergence and weak $γ$-convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.

math.OC↗
arXiv2013

Shape Optimization Problems for Metric Graphs

We consider the shape optimization problem $$\min\big\{{\mathcal E}(Γ)\ :\ Γ\in{\mathcal A},\ {\mathcal H}^1(Γ)=l\ \big\},$$ where ${\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\mathcal D}=\{D_1,\dots,D_k\}\subset{\mathbb R}^d$. The cost functional ${\mathcal E}(Γ)$ is the Dirichlet energy of $Γ$ defined through the Sobolev functions on $Γ$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

math.OC↗
arXiv2013

Lipschitz regularity of the eigenfunctions on optimal domains

We study the optimal sets $Ω^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(λ_1(Ω),\dots,λ_p(Ω)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues $λ_1(Ω),\dots,λ_p(Ω)$ of the Dirichlet Laplacian on $Ω$, a prototype being the problem $$ \min{\big\{λ_1(Ω)+\dots+ λ_p(Ω)\;:\;Ω\subset\mathbb{R}^d,\ |Ω|=1\big\}}. $$ We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots,u_p$ of the Dirichlet Laplacian on the optimal set $Ω^*$ and, as a corollary, we deduce that $Ω^*$ is open. For functionals depending only on a generic subset of the spectrum, as for example $λ_k(Ω)$ or $λ_{k_1}(Ω)+\dots+λ_{k_p}(Ω)$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.

math.AP↗
arXiv2013

Multiphase shape optimization problems

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\min\Big\{{g}(F_1(Ω_1),\dots,F_h(Ω_h))+ m\vert\,\bigcup_{i=1}^hΩ_i\vert :\ Ω_i\subset D,\ Ω_i\cap Ω_j =\emptyset\Big\},$ where $D\subset\mathcal{R}^d$ is a given bounded open set, $\vertΩ_i\vert$ is the Lebesgue measure of $Ω_i$ and $m$ is a positive constant. For a large class of such functionals, we analyse qualitative properties of the cells $Ω_i$ and the interaction between them. Each cell is itself subsolution for a (single phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e. $F_i=λ_{k_i}$.

math.AP↗
arXiv2013

Spectral optimization problems for potentials and measures

In the present paper we consider spectral optimization problems involving the Schrödinger operator $-Δ+μ$ on $\R^d$, the prototype being the minimization of the $k$ the eigenvalue $λ_k(μ)$. Here $μ$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\ds \int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in $\R^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.

math.OC↗
arXiv2013

Existence and regularity of minimizers for some spectral functionals with perimeter constraint

In this paper we prove that the shape optimization problem $$\min\left\{λ_k(Ω):\ Ω\subset\R^d,\ Ω \hbox{open},\ P(Ω)=1,\ |Ω|<+\infty\right\},$$ has a solution for any $k\in\N$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,α}$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f(λ_{k_1}(Ω),\dots,λ_{k_p}(Ω))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.

math.AP↗
arXiv2013

Faber-Krahn inequalities in sharp quantitative form

The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman. More generally, the result applies to every optimal Poincaré-Sobolev constant for the embeddings $W^{1,2}_0(Ω)\hookrightarrow L^q(Ω)$.

math.AP↗
arXiv2013

Optimal Potentials For Schrodinger Operators

We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

math.AP↗
arXiv2013

Some New Problems in Spectral Optimization

We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the {\it metric Laplacian}, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carathéodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schrödinger potential in suitable classes.

math.OC↗
arXiv2011

Shape Optimization Problems with Internal Constraint

We consider shape optimization problems with internal inclusion constraints, of the form $$\min\big\{J(Ω)\ :\ \Dr\subsetΩ\subset\R^d,\ |Ω|=m\big\},$$ where the set $\Dr$ is fixed, possibly unbounded, and $J$ depends on $Ω$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

math.AP↗