Alexandru Kristály

• Professor
• Babeș-Bolyai University

Inspired by Bakry [Lecture Notes in Math., 1994], we prove sharp hypercontractivity bounds of the heat flow $({\sf H}_t)_{t\geq 0}$ on ${\sf RCD}(0,N)$ metric measure spaces. The best constant in this estimate involves the asymptotic volume ratio, and its optimality is obtained by means of the sharp $L^2$-logarithmic Sobolev inequality on ${\sf RCD...

Our study deals with fine volume growth estimates on metric measures spaces supporting various Sobolev-type inequalities. We first establish the sharp Nash inequality on Riemannian manifolds with non-negative Ricci curvature and positive asymptotic volume ratio. The proof of the sharpness of this result provides a generic approach to establish a un...

The paper is devoted to prove Allard-Michael-Simon-type $L^p$-Sobolev $(p>1)$ inequalities with explicit constants on Euclidean submanifolds of any codimension. Such inequalities contain, beside the Dirichlet $p$-energy, a term involving the mean curvature of the submanifold. Our results require separate discussions for the cases $p\geq 2$ and $1<p...

The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $\Sigma$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $\Sigma$. It implies i...

The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenb...

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality...

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport , raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using $$L^1$$ L 1 -optimal transport approac...

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math. Ann., 2011). T...

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. T...

Recent results of three areas, pickup and delivery, optimal mass transportation, matching under preferences are highlighted. The topics themselves have been selected from the active research fields of Hungarian Operations Research. We also provide a short summary of selected research results from the 34th Hungarian Operations Research Conference, h...

In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math., 2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Mass Transportation, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. In this paper we affirmatively answer their questio...

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities....

We present a generic functional inequality on Riemannian manifolds, both in additive and multiplicative forms, that produces well known and genuinely new Hardy-type inequalities. For the additive version, we introduce Riccati pairs that extend Bessel pairs developed by Ghoussoub and Moradifam (Proc. Natl. Acad. Sci. USA, 2008 & Math.A nn., 2011). T...

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions...

Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. Combining the sharp $L^p$-log-Sobolev inequality with the Hamilton-Jacobi inequality, we establish a sharp hypercontractivity estimate for the...

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [Formula: see text]-dimensional Finsler manifolds having nonnegative [Formula: see text]-Ricci curvature. A byprod...

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 (Ashbaugh and Benguria in Duke Math J 78(1):1–17, 1995; Nadirashvili in A...

We prove Gagliardo-Nirenberg inequalities with three weights -- verifying a joint concavity condition -- on open convex cones of $\mathbb R^n$. If the weights are equal to each other the inequalities become sharp and we compute explicitly the sharp constants. For a certain range of parameters we can characterize the class of extremal functions; in...

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace-Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions...

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 negative...

We present the isometry between the 2-dimensional Funk model and the Finsler-Poincar\'e disk. Then, we introduce the Finslerian Poincar\'e 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.

By using optimal mass transport theory we prove a sharp isoperimetric inequality in CD(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf {CD}} (0,N)$$\end{do...

By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev ineq...

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce 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 Ch...

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 rev...

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 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-t...

Motivated by Nash equilibrium problems on ‘curved’ strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced by using nonsmooth analysis on Riemannian manifolds. Fixed point characterizations and existence of Nash-Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadama...

In the previous chapters elliptic problems are studied on bounded domains. In this chapter we treat various kinds of inequality problems on unbounded domains.

We study the weak solvability of a general m −dimensional (m = 2, 3) mechanical model describing the contact between a piezoelectric body and a conductive foundation. The piezoelectric effect is characterized by the coupling between the mechanical and the electrical properties of the materials. This coupling leads to the appearance of electric pote...

In this chapter we study various boundary value problems governed by the Laplace operator or a generalized Laplacian (p −Laplacian, variable exponent p(⋅) −Laplacian and Φ −Laplacian). The approach is variational, i.e., the solutions are the critical points of the corresponding energy functional defined on a suitable Sobolev space. Using the theore...

In this chapter we use fixed point and KKM theories in order to guarantee solutions for various differential inclusions and nonlinear hemivariational inequalities (both for scalar and vectorial cases).

Various classes of inequality problems driven by set-valued maps are investigated throughout this chapter via topological methods such as the Fan-KKM lemma, Tarafdar’s fixed point theorem or Mosco’s alternative. The standard approach is to establish the existence of at least one solution for the case of bounded closed convex constraint subsets of (...

Using the deformation lemmas from the previous chapter, various existence and multiplicity results concerning the critical points for nonsmooth functionals are provided in this chapter. We start with the Mountain Pass, Saddle Point and Linking theorems for locally Lipschitz functions which satisfy the generalized (φ − C)c condition. The “zero altit...

Deformation results are one of the main tools used in finding critical points. In this chapter we present several deformation results for nonsmooth functionals. First, two deformation results that require a generalized Cerami compactness condition are proved for locally Lipschitz functions defined on Banach spaces. Then, a deformation theorem for l...

In this chapter we briefly present basic properties of convex functions (as the Lipschitz property of convex functionals, the definition and main properties of the conjugate of a convex functional and the convex subdifferential), the direct method in the calculus of variations as well as the variational principle of Ekeland.

We consider a general mathematical model which describes the contact between a body and a rigid foundation, under the small deformations hypothesis. The behavior of the material is modeled by a monotone constitutive law, while on the potential contact zone nonmonotone boundary conditions are imposed. We propose a variational formulation in terms of...

We analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation, for static processes, under the small deformations hypothesis. Using the KKM lemma due to Fan (see Corollary D.1), we prove that the model has at least one weak solution. Moreover, we present several examples of constitutive laws and fri...

The main aspects in nonsmooth critical point theory are discussed throughout this chapter, namely the notion of critical point for functionals which are not differentiable, but are locally Lipschitz, a sum between C¹-functional and a convex l.s.c. functional or, more general, the sum between a locally Lipschitz and a convex l.s.c. functional. Vario...

In this chapter we focus our attention on the theory developed by Clarke for locally Lipschitz functionals. More precisely, we will investigate the properties of the generalized directional derivative and the Clarke subdifferential as well as the connection with the convex subdifferential. We also introduce two subdifferential notions for locally L...

By using a sharp isoperimetric inequality and an anisotropic symmetrization argument, we establish Morrey-Sobolev and Hardy-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature; in some cases we also discuss the sharpness of these functional inequalities. As applications, by using variational arguments, w...

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 rev...

We endow the disc D={(x1,x2)∈R2:x12+x22<4} with a Poincaré-type Randers metric Fλ, λ∈[0,1] that ’linearly’ interpolates between the usual Riemannian Poincaré disc model (λ=0, having constant sectional curvature -1 and zero S-curvature) and the Finsler–Poincaré metric (λ=1, having constant flag curvature -1/4 and constant S-curvature with isotropic...

This book provides a modern and comprehensive presentation of a wide variety of problems arising in nonlinear analysis, game theory, engineering, mathematical physics and contact mechanics. It includes recent achievements and puts them into the context of the existing literature. The volume is organized in four parts. Part I contains fundamental ma...

Let $(M,g)$ be a noncompact, complete $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and Euclidean volume growth, i.e., $0<$AVR$(g)\leq 1$, where AVR$(g)$ stands for the asymptotic volume ratio of $(M,g)$. The main purpose of the paper is to prove that the sharp Sobolev constants in various $L^p$-Sobolev inequalities on $(M,g)...

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 inequalit...

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...

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem (Dλ)−Δu(x)∈∂F(u(x))+λ∂G(u(x))inΩ;u≥0,inΩ;u=0,on∂Ω,where Ω⊂Rn is a bounded open domain, and ∂F and ∂G stand for the generalized gradients of the locally Lipschitz functions F and G. In this paper we provide a quite complete picture o...

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...

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥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 (n−1)416κ4 whenever the κ-Cartan-Hadamard conjec...

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 in...

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form \[\left(\int_E |u(x)|^q\,\omega(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,\sigma(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}\] where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E...

Motivated by mechanical problems where external forces are non-smooth, we consider the differential inclusion problem \[ \begin{cases} -\Delta u(x)\in \partial F(u(x))+\lambda \partial G(u(x))\ \mbox{in}\ \Omega \newline u\geq 0\ \mbox{in}\ \Omega \newline u= 0\ \mbox{on}\ \partial\Omega, \end{cases} \ \ \ \ \ \ \ \ \ \ \ \ {({\mathcal D}_\lambda)}...

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 $\textbf{K}\leq -\kappa^2$ for some $\kappa\geq 0.$ 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 $\fr...

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 $(\mathbb S^n,g_0)$. Moreo...

In this paper, we study the spectral problem in Finsler geometry. The spectrum of a Finsler metric measure manifold is defined to be the set of the critical values of the canonical energy functional, which is captured by a faithful dimension-like function. We estimate both the upper bound and the lower bound of such eigenvalues. Moreover, several f...

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 con...

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 ine...

This paper is devoted to investigate an interpolation inequality between the Brezis-V\'azquez and Poincar\'e 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...

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...

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...

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, McCa...

In this paper we prove Poincar\'e's lemma on some corank 1 sub-Riemannian structures of arbitrary dimension, solving partially an open problem formulated by Calin and Chang. Our proof is based on a Poincar\'e lemma stated on generic Riemannian manifolds and a suitable Ces\`aro-Volterra path integral formula established in local coordinates. As a by...

In this paper we prove the Poincar\'e 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 Poincar\'e lem...

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg–Pauli–Weyl uncertainty principle with the sharp constant in Rn (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 curvatur...

We prove that the fractional Yamabe equation $\mathcal L_\gamma u=|u|^\frac{4\gamma}{Q-2\gamma}u$ 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 $\mathcal L_\gamma$ denotes the CR fractional sub-Laplacian operator on $\mathb...

We prove that the fractional Yamabe equation $\mathcal L_\gamma u=|u|^\frac{4\gamma}{Q-2\gamma}u$ 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 $\mathcal L_\gamma$ denotes the CR fractional sub-Laplacian operator on $\mathb...

In this paper 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]. By using these inequalities together with variational methods and group-theoretical arguments, we also establi...

By using optimal mass transportation and a quantitative H\"older inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. As a first consequence, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities) are characterized in terms of the optimal...

By using optimal mass transportation and a quantitative H\"older 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 Pr\'ekopa-Leindler inequalities) are characterized in terms of the optimal transport...

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 Schmuckenschl\"ager. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space including the pr...

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 Schmuckenschl\"ager. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory...

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 \(K\ge 0\) and \(n\ge 2\), 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 i...

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 fr...

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...

Let (M, g) be an n−dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying ρ∆ g ρ ≥ n − 5 ≥ 0, where ∆ 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 se...

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $\rho\Delta_g \rho \geq n- 5\geq 0$, where $\Delta_g$ is the Laplace-Beltrami operator on $(M,g)$ and $\rho$ is the distance function from a given point. If $(M,g)$ supports a second-order Sobolev inequality with a constant $C>0$ close to...

In this Note, we present geodesic versions of the Borell–Brascamp–Lieb, Brunn–Minkowski and entropy inequalities on the Heisenberg group H n . Our arguments use the Riemannian approximation of H n combined with optimal mass-transportation techniques.

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, McCa...

We prove Sobolev-type interpolation inequalities on Hadamard manifolds and their optimality whenever the Cartan-Hadamard conjecture holds (e.g., in dimensions 2, 3 and 4). The existence of extremals leads to unexpected rigidity phenomena.

In this paper we study nonlinear Schr\"odinger-Maxwell systems on n−dimensional Hadamard manifolds, 3≤n≤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 sol...

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 Hadam...

In this paper we study the coupled Schr\"odinger-Maxwell system $$\left\{ \begin{array}{lll} -\triangle u+u +\phi u=\lambda \alpha(x) f(u)& {\rm in} & \mathbb R^3,\\ -\triangle \phi =u^2 & {\rm in} & \mathbb R^3, \end{array}\right. $$ where $\alpha\in L^\infty(\mathbb R^3)\cap L^{6/(5-q)}(\mathbb R^3)$ for some $q\in (0,1)$, and the continuous func...

In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 & {\rm on} & \partial\Omega, \end{array}\right.$$ where $\Omega\subset \m...

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 revers...

In this paper we study nonlinear Schr\"odinger-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-invari...

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 manif...

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 in...

In the present paper we prove a multiplicity result for an anisotropic sublinear elliptic problem with Dirichlet boundary condition, depending on a positive parameter λ. By variational arguments, we prove that for enough large values of λ, our anisotropic problem has at least two non-zero distinct solutions. In particular, we show that at least one...

In this paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. We first characterize the validity of Moser-Trudinger inequalities on complete non-compact $n-$dimensional Riemannian manifolds $(n\geq 2)$ with Ricci curvature bounded from below in terms of the volume growth of geodes...

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 manif...

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...

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...

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative...

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 sha...