We prove some inequalities of Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng-Yau operator.
In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.
In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.
Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del Pino and Dolbeault and the optimal L^2-Riemannian Gagliardo-Nirenberg inequality due to Broutteland in a unified framework.
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.
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-positive…
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 $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our r…
We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ \int_Ωu^{2γ-α-β}Δu^αΔu^βdx \geq c\int_Ω|Δu^γ|^2dx $, which seem to be of interest on their own right.
In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be pro…
We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we give a positive answer to an open question by A. Malchiodi on the existence of bounded Palais-Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.
In this paper we present a survey of the joint program with Fabrice Baudoin originated with the paper \cite{BG1}, and continued with the works \cite{BG2}, \cite{BBG}, \cite{BG3} and \cite{BBGM}, joint with Baudoin, Michel Bonnefont and Isidro Munive.
Let $(M,g)$ be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over $(M,g)$. We also derive an upper bound of such inequalities under the assumption that blowing up occur.
We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.
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 const…
This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.
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 …
In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy functional and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.
We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold $M$. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space $\mathbb{H}^n$.
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation and a sub-Riemannian Bonnet-Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distr…
This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequali…
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 Morr…
In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.
We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $Δ_{p}u := \Div(\abs{\nabla u}^{p-2}\nabla u)$. Namely, if $ρ$ is a nonnegative weight such that $-Δ_{p}ρ\geq0$, then the Hardy inequality $$c\int_{M}\frac{\abs{u}^{p}}{ρ^{p}}\abs{\nabla ρ}^{p} dv_{g} \leq \int_{M}\abs{\nabla u}^{p} dv_{g}, \quad u\in\Cinfinito_{0}(M)$$ holds. We show concrete examples specializing the function $ρ$.
In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $u$ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $u$.
We establish some sharp weighted trace inequalities $W^{1,2}(ρ^{1-2σ}, M)\hookrightarrow L^{\frac{2n}{n-2σ}}(\pa M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $ρ$ is a defining function of $M$ and $σ\in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.
We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As a byproduct, we characterize the cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex, answering to a question raised by Figalli and Rifford in [Geom. Funct. Anal. (2010) 20: 124]. As an application, we deduce sharp and intrinsic Borell-…
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 2$. This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy inequality \[ {\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} + B_{opt}\right) \] for all $u \in BV(M)$ with $\|u\|_{L^1(M)} = 1$ and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent a optimal $L^1$-Sobolev inequality obtained by Druet [6].
We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
Let $(Σ,g)$ be a closed Riemannian surface, $W^{1,2}(Σ,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(Σ,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in W^{1,2}(Σ,g)$ with $\int_Σudv_g=0$ and $u(σ(x))=u(x)$ for all $σ\in \textbf{G}$ and all $x\inΣ$. Denote the number of distinct points of the set $\{σ(x): σ\in \textbf{G}\}$ by $I(x)$ and $\ell=\inf_{x\in Σ}I(x)$. Let $λ_1^\textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $\mathscr{H}_\textbf{G}$. Using blow-up analysis, we prove that if $α<…
Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $Γ^Z$ and we assume that $Γ$ and $Γ^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequa…
We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and sharp Euclidean isoperimetric inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green's functional using the asymptotic behavior of the Lorentz norm deficit of Green's function at the infinity, as well as the harmonic radius of a regular …
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 $\mathbb H^n$ develo…
We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.
