arXiv subjects

Dorin Bucur

Publications and source records attributed to Dorin Bucur.

At least 37 records · Page 2Linked to original sources

arXiv2023

Mean-to-max ratio of the torsion function and honeycomb structures

In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.

math.AP↗

arXiv2023

A free discontinuity approach to optimal profiles in Stokes flows

In this paper we study obstacles immerged in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.

math.AP↗

arXiv2022

Sharp inequalities for Neumann eigenvalues on the sphere

We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaugh-Benguria result can be relaxed to a weaker one, namely empty intersection with a geodesic ball of the prescribed volume. Although we do not prove that this last inclusion result is sharp, for a mass less than the half of the sphere, we numerically identify a density with higher first eigenvalue than the corresponding geodesic ball and with support equal to the full sphere $\mathbb{S}^2$.

math.AP↗

arXiv2022

Shape optimization of a thermal insulation problem

We study a shape optimization problem involving a solid $K\subset\mathbb{R}^n$ that is maintained at constant temperature and is enveloped by a layer of insulating material $Ω$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $(K,Ω)$ with prescribed measure for $K$ and $Ω$, and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $\partialΩ$) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

math.AP↗

arXiv2022

Boundary behavior of Robin problems in non-smooth domains

We analyze strict positivity at the boundary for nonnegative solutions of Robin problems in general (non-smooth) domains, e.g. open sets with rectifiable topological boundaries having finite Hausdorff measure. This question was raised by Bass, Burdzy and Chen in 2008 for harmonic functions, in a probabilistic context. We give geometric conditions such that the solutions of Robin problems associated to general elliptic operators of $p$-Laplacian type, with a positive right hand side, are globally or locally bounded away from zero at the boundary. Our method, of variational type, relies on the analysis of an isoperimetric profile of the set and provides quantitative estimates as well.

math.AP↗

arXiv2022

Maximization of Neumann eigenvalues

This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of $k$ equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in ${\mathbb R}$. Based on the relaxed formulation, we provide numerical approximations of optimal densities for $k=1, \dots, 8$ in ${\mathbb R}^2$.

math.AP↗

arXiv2022

On the Polygonal Faber-Krahn Inequality

It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with $n$ sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each $n \ge 5$ the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For $n=5, 6,7, 8$ we perform this computation and certify the numerical approximation by finite elements, up to machine errors.

math.OC↗

arXiv2022

Alexandrov theorem for general nonlocal curvatures: the geometric impact of the kernel

For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability" assumption on $h$, we prove that these sets are finite unions of equal balls, as soon as they satisfy a natural nondegeneracy condition. Both the radius of the balls and their mutual distance can be controlled from below in terms of suitable parameters depending explicitly on the measure of the level sets of $h$. In the simplest, common case, in which $h$ is positive, bounded and decreasing, our result implies that any bounded open set or any bounded measurable set with finite perimeter which has constant nonlocal $h$-mean curvature has to be a ball.

math.DG↗

arXiv2021

Rigidity for measurable sets

Let $Ω\subset \mathbb{R}^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $Ω\cap B_r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of $Ω$. We prove that $Ω$ is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than $r$ which is either open and connected, or of finite perimeter and indecomposable. The proof requires reinventing each step of the moving planes method by Alexandrov in the framework of measurable sets.

math.MG↗

arXiv2020

Degenerate free discontinuity problems and spectral inequalities in quantitative form

We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with negative sign. This is motivated by searching quantitative inequalities for best constants of Sobolev-Poincaré inequalities with trace terms in $\mathbb{R}^n$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.

math.AP↗

arXiv2020

Asymptotic behaviour of the Steklov problem on dumbbell domains

We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.

math.AP↗

arXiv2020

Stability and instability issues of the Weinstock inequality

Given two planar, conformal, smooth open sets $Ω$ and $ω$, we prove the existence of a sequence of smooth sets $Ω_n$ which geometrically converges to $Ω$ and such that the (perimeter normalized) Steklov eigenvalues of $Ω_n$ converge to the ones of $ω$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.

math.AP↗

arXiv2020

Sign changing solutions of Poisson's equation

Let $Ω$ be an open, possibly unbounded, set in Euclidean space $\R^m$ with boundary $\partialΩ,$ let $A$ be a measurable subset of $Ω$ with measure $|A|$, and let $γ\in (0,1)$. We investigate whether the solution $v_{\Om,A,γ}$ of $-Δv=γ{\bf 1}_{Ω\setminus A}-(1-γ){\bf 1}_{A}$ with $v=0$ on $\partial Ω$ changes sign. Bounds are obtained for $|A|$ in terms of geometric characteristics of $\Om$ (bottom of the spectrum of the Dirichlet Laplacian, torsion, measure, or $R$-smoothness of the boundary) such that ${\rm essinf} v_{\Om,A,γ}\ge 0$. We show that ${\rm essinf} v_{\Om,A,γ}<0$ for any measurable set $A$, provided $|A| >γ|\Om|$. This value is sharp. We also study the shape optimisation problem of the optimal location of $A$ (with prescribed measure) which minimises the essential infimum of $v_{\Om,A,γ}$. Surprisingly, if $\Om$ is a ball, a symmetry breaking phenomenon occurs.

math.AP↗

arXiv2018

Optimal partitions for Robin Laplacian eigenvalues

We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of $k$ mutually disjoint {\it open} sets which have a $\mathcal H ^ {d-1}$-countably rectifiable boundary and are contained into a given box $D$ in $R^d$

math.AP↗

arXiv2018

Maximization of the second non-trivial Neumann eigenvalue

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{ó}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

math.AP↗

arXiv2017

Weinstock inequality in higher dimensions

We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n\ge 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultanously the surface area, the volume and the boundary momentum of convex sets. As a by product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue

math.AP↗

arXiv2017

Two optimization problems in thermal insulation

We consider two optimization problems in thermal insulation: in both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The total mass of the insulating material is prescribed.. The first problem deals with the case in which a given heat source is present, while in the second one there are no heat sources and the goal is to have the slowest decay of the temperature. In both cases an optimal distribution of the insulator around the thermal body exists; when the body has a circular symmetry, in the first case a constant heat source gives a constant thickness as the optimal solution, while surprisingly this is not the case in the second problem, where the circular symmetry of the optimal insulating layer depends on the total quantity of insulator at our disposal. A symmetry breaking occurs when this total quantity is below a certain threshold. Some numerical computations are also provided, together with a list of open questions.

math.OC↗

arXiv2017

On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity.

math.OC↗