arXiv subjects

Dorin Bucur

Publications and source records attributed to Dorin Bucur.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds $μ_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $Ω\subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_Ω^2 μ_k(Ω) \leq μ_{k,d}^* - C(k,d) a_2(Ω)^2/D_Ω^2$$ where $D_Ω$ is the diameter of $Ω$ and $a_2(Ω)$ is the second largest semiaxis of the John ellipsoid of $Ω$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.

math.AP↗

arXiv2026

Symmetry breaking in the polygonal Szegö-Weinberger inequality as $p\to1^+$: the longest shortest-fence quadrilateral

We consider Pólya's problem of finding, among convex sets of prescribed area, the one with the longest shortest fence, in the polygonal setting, namely when the class of competitors is restricted to polygons with a prescribed number of sides. While it is straightforward to show that, among triangles, the optimal shape is the equilateral one, we prove that symmetry breaking occurs in the case of quadrilaterals: the optimal quadrilateral is not the square. More precisely, we identify it as a specific isosceles trapezium, which is uniquely determined, up to homotheties and rigid motions, by an elementary equation for its base angle. The proof combines analytical arguments and rigorous interval-arithmetic computations.

math.OC↗

arXiv2026

Concavity and hot spots in elliptic problems under mixed boundary conditions

We consider the torsion function and the first Laplacian eigenfunction in a convex curvilinear sector in the plane, under homogeneous Neumann conditions on the two straight lateral sides and a homogeneous Dirichlet condition on the remaining part of the boundary. We prove that they are, respectively, strictly $(\frac 1 2)$-concave and strictly log-concave, provided the interior angles at the vertices are at most $\frac π2$. Under the same assumption, we further establish a billiard-type concavity, obtained through reflections across the Neumann sides. As a consequence, we deduce that there exists a unique hot spot, located at the Neumann--Neumann vertex, and that some monotonicity properties hold along suitable segments. Finally, we prove that the associated variational energies, namely the mixed torsional rigidity and the first mixed Laplacian eigenvalue, satisfy Brunn--Minkowski type inequalities in the class of convex curvilinear sectors with a fixed opening angle.

math.AP↗

arXiv2025

Sharp Quantitative Stability of the Dirichlet spectrum near the ball

Let $Ω\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $λ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator of $Ω$ with Dirichlet boundary conditions on $\partialΩ$. In this work, we answer the following question: if $λ_1(Ω)-λ_1(B)$ is small, how large can $|λ_k(Ω)-λ_k(B)|$ be ? We establish quantitative bounds of the form $|λ_k(Ω)-λ_k(B)|\le C (λ_1(Ω)-λ_1(B))^α$ with sharp exponents $α$ depending on the multiplicity of $λ_k(B)$. We first show that such an inequality is valid with $α=1/2$ for any $k$, improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent $α=1$ if $λ_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $λ_{k}(B)$ is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as the minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler-Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.

math.AP↗

arXiv2025

Optimal domains for the Cheeger inequality

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(Ω)=\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\le q<p\le+\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\partialΩ$. We study both the maximization and minimization problems for ${\mathcal{F}}_{p,q}$, and show the existence of optimal domains in ${\mathbb{R}}^d$, along with some of their qualitative properties. Surprisingly, the case of a bounded box $D$ constraint $$\max\Big\{λ_q(Ω)\ :\ Ω\subset D,\ λ_p(Ω)=1\Big\},$$ leads to a problem of different nature, for which the existence of a solution is shown by analyzing optimal capacitary measures. In the last section we list some interesting questions that, in our opinion, deserve to be investigated.

math.OC↗

arXiv2025

On Poincaré constants related to isoperimetric problems in convex bodies

For any convex set $Ω\subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincaré constant in $W ^ {1, 1}(Ω)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an extra term giving account for the flatness of the domain. In dimension $N = 2$, we are able to make the extra term completely explicit, thus providing a new Bonnesen-type inequality for the Poincaré constant in terms of diameter and inradius. Such estimate is sharp, and it is asymptotically attained when the domain is the intersection of a ball with a strip bounded by parallel straight lines, symmetric about the centre of the ball. As a key intermediate step, we prove that the ball maximizes the Poincaré constant in $W ^ {1, 1} (Ω)$, among convex bodies $Ω$ of given constant width.

math.AP↗

arXiv2025

Spherical caps do not always maximize Neumann eigenvalues on the sphere

We prove the existence of an open set $Ω\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.

math.AP↗

arXiv2025

The geometric size of the fundamental gap

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, our main result strengthens Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions à la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schrödinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in $\mathbb{R}^N$, thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang.

math.SP↗

arXiv2025

On localisation of eigenfunctions of the Laplace operator

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions $κ$-localise in $L^2$.

math.SP↗

arXiv2025

Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates

We prove a fourth order analogue of the Saint-Venant inequality: the mean deflection of a clamped plate under uniform transverse load is maximal for the ball, among plates of prescribed volume in any dimension of space. The method works in Euclidean space, hyperbolic space, and the sphere. Similar results for clamped plates under small compression and for the compliance under non-uniform loads are proved to hold in two dimensional Euclidean space, with the higher dimensional and curved cases of those problems left open.

math.AP↗

arXiv2024

On the torsion function for simply connected, open sets in $\R^2$

For an open set $\Om \subset \R^2$ let $λ(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown that there exist {$η_1>0,η_2>0$} such that { (i) $\|w_{\Om}\|_{\infty} λ(\Om)\ge 1+η_1$ for any non-empty, open, simply connected set $\Om\subset \R^2$ with $\lb(\Om) >0$, (ii) $\|w_{\Om}\|_1λ(\Om)\le {(1-η_2)}|\Om|$ for any non-empty, open, simply connected set $\Om\subset\R^2$ with finite measure $|\Om|$.

math.SP↗

arXiv2024

A sharp quantitative nonlinear Poincaré inequality on convex domains

For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $μ_ p (Ω, φ)$ of the $p$-Laplacian on a convex domain $Ω\subset \mathbb{R}^N$ with a power-concave weight $φ$. Our result improves the classical estimate in terms of the diameter, first stated in a seminal paper by Payne and Weinberger: we add in the lower bound an extra term depending on the second largest John semi-axis of $Ω$ (equivalent to a power of the width in the special case $N = 2$). The power exponent in the extra term is sharp, and the constant in front of it is explicitly tracked, thus enlightening the interplay between space dimension, nonlinearity and power-concavity. Moreover, we attack the stability question: we prove that, if $μ_ p (Ω, φ)$ is close to the lower bound, then $Ω$ is close to a thin cylinder, and $φ$ is close to a function which is constant along its axis. As intermediate results, we establish a sharp $L^ \infty$ estimate for the associated eigenfunctions, and we determine the asymptotic behaviour of $μ_ p (Ω, φ)$ for varying weights and domains, including the case of collapsing geometries.

math.AP↗

arXiv2024

Polygonal Faber-Krahn inequality: Local minimality via validated computing

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb R^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for $n \in \{5,6\}$. All computations, including the finite element computations, are realized using interval arithmetic.

math.NA↗

arXiv2023

The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities

Given a non-increasing and radially symmetric kernel in $L ^ 1 _{\rm loc} (\Bbb{R} ^ 2 ; \Bbb{R}_+)$, we investigate counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with given area and $N$ sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. We prove that, for every $N \geq 3$, the regular $N$-gon is optimal for Hardy-Littlewood inequality. Things go differently for Riesz inequality: while for $N = 3$ and $N = 4$ it is known that the regular triangle and the square are optimal, for $N\geq 5$ we prove that symmetry or symmetry breaking may occur (i.e. the regular $N$-gon may be optimal or not), depending on the value of $N$ and on the choice of the kernel.

math.OC↗

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↗