arXiv subjects

Alexandru Kristály

Publications and source records attributed to Alexandru Kristály.

At least 55 records · Page 3Linked to original sources

arXiv2018

Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.

math.AP↗

arXiv2017

Nodal solutions for the fractional Yamabe problem on Heisenberg groups

We prove that the fractional Yamabe equation $\mathcal L_γu=|u|^\frac{4γ}{Q-2γ}u$ on the Heisenberg group $\mathbb H^n$ has $[\frac{n+1}{2}]$ sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where $\mathcal L_γ$ denotes the CR fractional sub-Laplacian operator on $\mathbb H^n$, $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$, and $γ\in \bigcup_{k=1}^n[k,\frac{kQ}{Q-1})$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere $S^{2n+1}$ combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group ${\bf U}(n+1).$

math.AP↗

arXiv2017

Poincaré's lemma on some non-Euclidean structures

In this paper we prove the Poincaré lemma on some $n$-dimensional corank 1 sub-Riemannian structures, formulating the $\frac{(n-1)n(n^2+3n-2)}{8}$ necessarily and sufficiently 'curl-vanishing' compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. Our proof is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.

math.AP↗

arXiv2017

Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $ρΔ_g ρ\geq n- 5\geq 0$, where $Δ_g$ is the Laplace-Beltrami operator on $(M,g)$ and $ρ$ is the distance function from a given point. If $(M,g)$ supports a second-order Sobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ in the second-order Sobolev inequality in $\mathbb R^n$, we show that a global volume non-collapsing property holds on $(M,g)$. The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-order Sobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ is isometric to the Euclidean space $\mathbb R^n$.

math.AP↗

arXiv2017

Multipolar Hardy inequalities on Riemannian manifolds

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

math.AP↗

arXiv2017

Sharp uncertainty principles on Riemannian manifolds: the influence of curvature

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb R^n$ (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows: (a) When $(M,g)$ has non-positive sectional curvature, the sharp HPW principle holds on $(M,g)$. However, positive extremals exist in the sharp HPW principle if and only if $(M,g)$ is isometric to $\mathbb R^n$, $n={\rm dim}(M)$. (b) When $(M,g)$ has non-negative Ricci curvature, the sharp HPW principle holds on $(M,g)$ if and only if $(M,g)$ is isometric to $\mathbb R^n$. Since the sharp HPW principle and the Hardy-Poincaré inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds.

math.AP↗

arXiv2016

Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities

Let $({M},\textsf{d},\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\textsf{CD}(K,n)$ for some $K\geq 0$ and $n\geq 2$, and a lower $n-$density assumption at some point of $M$. We prove that if $({M},\textsf{d},\textsf{m})$ supports the Gagliardo-Nirenberg inequality or any of its limit cases ($L^p-$logarithmic Sobolev inequality or Faber-Krahn-type inequality), then a global non-collapsing $n-$dimensional volume growth holds, i.e., there exists a universal constant $C_0>0$ such that $\textsf{m}( B_x(ρ))\geq C_0 ρ^n$ for all $x\in {M}$ and $ρ\geq 0,$ where $B_x(ρ)=\{y\in M:{\sf d}(x,y)<ρ\}$. Due to the quantitative character of the volume growth estimate, we establish several rigidity results on Riemannian manifolds with non-negative Ricci curvature supporting Gagliardo-Nirenberg inequalities by exploring a quantitative Perelman-type homotopy construction developed by Munn (J. Geom. Anal., 2010). Further rigidity results are also presented on some reversible Finsler manifolds.

math.AP↗

arXiv2016

Metric projections versus non-positive curvature

In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results are also established on reversible/non-reversible Finsler-Minkowski spaces.

math.DG↗

arXiv2016

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for $H-$convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous $H-$convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

math.AP↗

arXiv2016

Nash-type equilibria on Riemannian manifolds: a variational approach

Motivated by Nash equilibrium problems on 'curved' strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash-Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifoleds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the {non-positivity} of the sectional curvature of complete and simply connected Riemannian spaces, delimiting the Hadamard manifolds as the optimal geometrical framework of Nash-Stampacchia equilibrium problems. Our analytical approach exploits various elements from set-valued and variational analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.

math.AP↗

arXiv2016

Multiple solutions for a Neumann system involving subquadratic nonlinearities

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -Δu+a(x)u=λc(x) F_u(u,v)& {\rm in} & Ω,\\ -Δv+b(x)v=λc(x) F_v(u,v)& {\rm in} & Ω,\\ \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0 & {\rm on} & \partialΩ, \end{array}\right.$$ where $Ω\subset \mathbb R^N$ is a smooth open bounded domain, $ν$ denotes the outward unit normal to $\partial Ω$, $λ\geq 0$ is a parameter, $a,b,c\in L_+^\infty(Ω)\setminus\{0\},$ and $F\in C^1(\mathbb{R}^2,\mathbb{R})\setminus\{0\}$ is a nonnegative function which is subquadratic at infinity. Two nearby numbers are determined in explicit forms, $\underline λ$ and $\overline λ$ with $ 0<\underlineλ\leq \overline λ$, such that for every $0\leq λ<\underline λ$, system $(N_λ)$ has only the trivial pair of solution, while for every $λ>\overline λ$, system $(N_λ)$ has at least two distinct nonzero pairs of solutions.

math.AP↗

arXiv2016

On the Schrödinger-Maxwell system involving sublinear terms

In this paper we study the coupled Schrödinger-Maxwell system $$\left\{ \begin{array}{lll} -\triangle u+u +ϕu=λα(x) f(u)& {\rm in} & \mathbb R^3,\\ -\triangle ϕ=u^2 & {\rm in} & \mathbb R^3, \end{array}\right. $$ where $α\in L^\infty(\mathbb R^3)\cap L^{6/(5-q)}(\mathbb R^3)$ for some $q\in (0,1)$, and the continuous function $f:\mathbb{R}\to\mathbb{R}$ is superlinear at zero and sublinear at infinity, e.g., $f(s) = \min(|s|^r,|s|^p)$ with $0 0$, non-existence and multiplicity results are obtained.

math.AP↗

arXiv2016

Schrödinger-Maxwell systems on Hadamard manifolds

In this paper we study nonlinear Schrödinger-Maxwell systems on $n-$dimensional Hadamard manifolds, $3\leq n\leq 5.$ The main difficulty resides in the lack of compactness of such manifolds which is recovered by exploring suitable isometric actions. By combining variational arguments, some existence, uniqueness and multiplicity of isometry-invariant weak solutions are established for the Schrödinger-Maxwell system depending on the behavior of the nonlinear term.

math.AP↗

arXiv2016

Singular Poisson equations on Finsler-Hadamard manifolds

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds having finite reversibility constant.

math.AP↗

arXiv2016

Quantitative Rellich Inequalities on Finsler-Hadamard Manifolds

In this paper we are dealing with quantitative Rellich inequalities on Finsler-Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants are also studied. Our results complement those of Yang, Su and Kong [Commun. Contemp. Math., 2014].

math.AP↗

arXiv2014

Elliptic problems on the ball endowed with Funk-type metrics

We study Sobolev spaces on the $n-$dimensional unit ball $B^n(1)$ endowed with a parameter-depending Finsler metric $F_a$, $a\in [0,1],$ which interpolates between the Klein metric $(a=0)$ and Funk metric $(a=1)$, respectively. We show that the standard Sobolev space defined on the Finsler manifold $(B^n(1),F_a)$ is a vector space if and only if $a\in [0,1).$ Furthermore, by exploiting variational arguments, we provide non-existence and existence results for sublinear elliptic problems on $(B^n(1),F_a)$ involving the Finsler-Laplace operator whenever $a\in [0,1).$

math.AP↗

arXiv2014

Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

Two Morrey-Sobolev inequalities (with support-bound and $L^1-$bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in $\mathbb R^n$. We prove the following results in both cases: $\bullet$ If $(M,g)$ is a {\it Cartan-Hadamard manifold} which verifies the $n-$dimensional Cartan-Hadamard conjecture, sharp Morrey-Sobolev inequalities hold on $(M,g)$. Moreover, extremals exist if and only if $(M,g)$ is isometric to the standard Euclidean space $(\mathbb R^n,e)$. $\bullet$ If $(M,g)$ has {\it non-negative Ricci curvature}, $(M,g)$ supports the sharp Morrey-Sobolev inequalities if and only if $(M,g)$ is isometric to $(\mathbb R^n,e)$.

math.AP↗

arXiv2014

What do `convexities' imply on Hadamard manifolds?

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well known Euclidean counterparts.

math.DG↗