arXiv subjects

Alexandru Kristály

Publications and source records attributed to Alexandru Kristály.

At least 37 records · Page 2Linked to original sources

arXiv2022

Lord Rayleigh's Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

We affirmatively solve the analogue of Lord Rayleigh's conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (M. Ashbaugh & R. Benguria, 1995, and N. Nadirashvili, 1995) and negatively curved (A. Kristály, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh's conjecture in higher dimensions. The proofs rely on an Ashbaugh-Benguria-Nadirashvili-Talenti nodal-decomposition argument, on the Lévy-Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case - when the Ricci curvature is non-negative - we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.

math.DG↗

arXiv2022

Three isometrically equivalent models of the Finsler-Poincaré disk

We present the isometry between the 2-dimensional Funk model and the Finsler-Poincaré disk. Then, we introduce the Finslerian Poincaré upper half plane model, which turns out to be also isometrically equivalent to the previous models. As application, we state the gapless character of the first eigenvalue for the aforementioned three spaces.

math.DG↗

arXiv2022

Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of S. Brendle (2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we use appropriate symmetrization techniques and optimal volume non-collapsing properties. The equality cases in the latter inequalities are also characterized by stating that sufficiently smooth, nonzero extremal functions exist if and only if the Riemannian manifold is isometric to the Euclidean space.

math.DG↗

arXiv2021

On the geometry of irreversible metric-measure spaces: Convergence, Stability and Analytic aspects

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable -- reversibility depending -- Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.

math.DG↗

arXiv2020

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

math.AP↗

arXiv2020

A bipolar Hardy inequality on Finsler manifolds

We establish a bipolar Hardy inequality on complete, not necessarily reversible Finsler manifolds. We show that our result strongly depends on the geometry of the Finsler structure, namely on the reversibility constant $r_F$ and the uniformity constant $l_F$. Our result represents a Finslerian counterpart of the Euclidean multipolar Hardy inequality due to Cazacu and Zuazua (2013) and the Riemannian case considered by Faraci, Farkas and Kristály (2018).

math.DG↗

arXiv2020

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,ω(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,σ(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $ω,σ:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, ω, σ) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $ω$ and $σ$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.

math.AP↗

arXiv2020

Sharp uncertainty principles on general Finsler manifolds

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

math.AP↗

arXiv2020

Fundamental tones of clamped plates in nonpositively curved spaces

We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\frac{(n-1)^4}{16}κ^4$ whenever the $κ$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in $(M,g)$ of volume $v>0$ is not less than the corresponding fundamental tone of a geodesic ball of the same volume $v$ in the space of constant curvature $-κ^2$ provided that $v\leq c_n/κ^n$ with $c_2\approx 21.031$ and $c_3\approx 1.721$, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. $\textbf{K}\equivκ=0$). The sharpness of our results requires the validity of the $κ$-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on $(M,g)$) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

math.AP↗

arXiv2020

Differential inclusions involving oscillatory terms

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -Δu(x)\in \partial F(u(x))+λ\partial G(u(x))\ \mbox{in}\ Ω\newline u\geq 0\ \mbox{in}\ Ω\newline u= 0\ \mbox{on}\ \partialΩ, \end{cases} \ \ \ \ \ \ \ \ \ \ \ \ {({\mathcal D}_λ)} \] where $Ω\subset {\mathbb R}^n$ is a bounded open domain, and $\partial F$ and $\partial G$ stand for the generalized gradients of the locally Lipschitz functions $F$ and $G$. In this paper we provide a quite complete picture on the number of solutions of $({\mathcal D}_λ)$ whenever $\partial F$ oscillates near the origin/infinity and $\partial G$ is a generic perturbation of order $p>0$ at the origin/infinity, respectively. Our results extend in several aspects those of Kristály and Moroşanu [J. Math. Pures Appl., 2010].

math.AP↗

arXiv2019

Nonlinear spectrums of Finsler manifolds

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce \textit{faithful dimension pairs} by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik-Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry-Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik-Schnirelmann dimension pair.

math.DG↗

arXiv2019

Topological rigidity of compact manifolds supporting Sobolev-type inequalities

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere $(\mathbb S^n,g_0)$, we prove that $(M,g)$ is topologically close to $(\mathbb S^n,g_0)$. Moreover, the Sobolev constants on $(M,g)$ are precisely the optimal constants on the sphere $(\mathbb S^n,g_0)$ if and only if $(M,g)$ is isometric to $(\mathbb S^n,g_0)$; in particular, the latter result answers a question of V.H. Nguyen.

math.AP↗

arXiv2019

Jacobian determinant inequality on corank 1 Carnot groups with applications

We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschläger. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups.

math.AP↗

arXiv2019

New geometric aspects of Moser-Trudinger inequalities on Riemannian manifolds: the non-compact case

In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser-Trudinger inequalities on complete non-compact $n-$dimensional Riemannian manifolds $(n\geq 2)$ with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the $n-$Laplace-Beltrami operator and a critical nonlinearity on $n-$dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].

math.AP↗

arXiv2018

Interpolation between Brezis-Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis-Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler-Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.

math.AP↗

arXiv2018

Energy properties of critical Kirchhoff problems with applications

In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the parameters for the convexity and validity of the Palais-Smale condition of the same energy functional. We then apply these regularity properties in the study of some elliptic problems involving the critical Kirchhoff term.

math.AP↗

arXiv2018

New features of the first eigenvalue on negatively curved spaces

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model $n$-dimensional hyperbolic space, complementing the results of Borisov and Freitas (\textit{Comm. Anal. Geom.} 25: 507--544, 2017). We then give a synthetic proof of Cheng's sharp eigenvalue comparison theorem in metric measure spaces satisfying a 'negatively curved' Bishop-Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (\textit{J. Differential Geom.} 4: 359--366, 1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

math.AP↗

arXiv2018

Equality in Borell-Brascamp-Lieb inequalities on curved spaces

By using optimal mass transportation and a quantitative Hölder inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Prékopa-Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge-Ampère equation, we {give a new proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb inequalities in the Euclidean setting. When the $n$-dimensional Riemannian manifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, it turns out that equality in the Borell-Brascamp-Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature $k$. A precise characterization is provided for the equality in the Lott-Sturm-Villani-type distorted Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.

math.AP↗