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Saikat Mazumdar

Publications and source records attributed to Saikat Mazumdar.

14 recordsLinked to original sources
arXiv2026

Compactness of conformal metrics with constant $Q$-curvature of higher order

Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of metrics conformal to $g$ with prescribed constant positive $Q$-curvature of order $2k$ --- or, equivalently, of the set of positive solutions for the $2k$-th order $Q$-curvature equation. Under a natural positivity-preserving condition on $P_{g}$ we establish compactness, for an arbitrary $1 \le k < \frac{n}{2}$, under different assumptions: $(M,g)$ is locally conformally flat and $P_g$ has positive mass in $M$; $2k+1 \le n \le 2k+5$ and $P_g$ has positive mass in $M$, whenever the mass is defined; $n \ge 2k+4$ and the Weyl tensor $\text{Weyl}_g$ never vanishes in $M$. For an arbitrary $1 \le k < \frac{n}{2}$ the expression of $P_g$ is not explicit, which is an obstacle to proving compactness. We overcome this by relying on Juhl's recursive formulae for $P_g$ to perform a refined blow-up analysis for solutions of the $Q$-curvature equation and to prove a Weyl vanishing result for $P_g$. Our result also hints that the threshold dimension for compactness for the $2k$-th order $Q$-curvature equation diverges as $k \to + \infty$.

math.AP↗
arXiv2026

Barycenter technique for the higher order $Q$-curvature equation

Let $k\ge1$ be an integer, and $(M,g)$ be a smooth, closed Riemannian manifold of dimension $2k+1\le n\le 2k+3$, or $(M,g)$ be locally conformally flat of dimension $n\ge 2k+1$. Applying the Bahri-Coron barycenter method, we show the existence of a conformal metric with constant $Q$-curvature of order $2k$, or equivalently, the existence of a positive solution for the $2k$-th order $Q$-curvature equation involving the GJMS operator $P_{g}$. We only assume a natural positivity preserving condition on $P_{g}$ and do not suppose any condition on the sign of the {\emph{mass}} of $P_{g}$. In particular, we obtain existence without using a positive mass theorem.

math.DG↗
arXiv2025

Compactness for the Hardy-Sobolev equation on manifolds

Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$, and let $h \in C^1(M)$ be such that the operator $Δ_g + h$ is coercive. Fix $x_0 \in M$ and $s \in (0, 2)$. We obtain uniform bounds on the solutions of the critical \emph{Hardy-Sobolev equation}: \begin{equation}\label{HS0} \tag{{\color{MainRed}HS}} \left\{\begin{array}{ll} Δ_{g}u + hu = \frac{u^{\crits-1}}{d_{g}(\xo,x)^{s}} & \hbox{ in }M\setminus\{\xo\}, \\ \qquad u > 0 &\hbox{ in }M\setminus\{\xo\}, \end{array}\right. \end{equation} where $Δ_{g}:=-\diver_{g}(\nabla)$ and $\crits:=2(n-s)/(n-2)$. More precisely, we assume $h(x_0)<\frac{(n-2)(6-s)}{12(2n-2-s)}\mathrm{Scal}_g(x_0),$ when $n \geq 4$, and $h\le\frac{1}{8}\sg$, $h(\xo)<\frac{1}{8}\sg(\xo)$ when $n = 3$. Here, $\mathrm{Scal}_g$ denotes the scalar curvature of $(M, g)$. These conditions were introduced in \cite{HCA4}, and shown to be optimal in \cite{CAR} for a single bubble configuration when $n\ge7$ . \noindent We do not assume any bounds on the energy or the Sobolev norm of the solutions.

math.AP↗
arXiv2023

Sharp quantitative stability of Struwe's decomposition of the Poincaré-Sobolev inequalities on the hyperbolic space: Part I

A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincaré-Sobolev equation on the hyperbolic space $$ -Δ_{\mathbb{B}^n} u-λu = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n), $$ are unique up to hyperbolic isometries where $n \geq 3,$ $1 < p \leq \frac{n+2}{n-2} $ and $λ\leq \frac{(n-1)^2}{4}.$ We prove under certain bounds on $\|\nabla u \|_{L^2(\mathbb{B}^n)}$ the inequality $$ δ(u) \lesssim \|Δ_{\mathbb{B}^n} u+ λu + u^{p}\|_{H^{-1}}, $$ holds whenever $p >2$ and hence forcing the dimensional restriction $3 \leq n \leq 5,$ where $δ(u)$ denotes the $H^1$ distance of $u$ from the manifold of sums of hyperbolic bubbles. Moreover, it fails for any $n \geq 3$ and $p \in (1,2].$ This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent $p$ is $>2$ or $\leq 2$. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration. Mech. Anal, 237 (2020)] but we notice a striking dependence on the exponent $p$ in the subcritical regime as well which is not present in the flat case. Our technique is an amalgamation of Figalli and Glaudo's method and builds upon a series of new and novel estimates on the interaction of hyperbolic bubbles and their derivatives and improved eigenfunction integrability estimates. Since the conformal group coincides with the isometry group of the hyperbolic space, we perceive a remarkable distinction in arguments and techniques to achieve our main results compared to that of the Euclidean case.

math.AP↗
arXiv2022

Existence results for the higher-order $Q$-curvature equation

We obtain existence results for the $Q$-curvature equation of order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an integer. We obtain these results under the assumptions that the Yamabe invariant of order $2k$ is positive and the Green's function of the corresponding operator is positive, which are satisfied for instance when the manifold is Einstein with positive scalar curvature. In the case where $2k+1\le n\le2k+3$ or $(M,g)$ is locally conformally flat, we assume moreover that the operator has positive mass. In the case where $n\ge2k+4$ and $(M,g)$ is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on $n$ and $k$.

math.AP↗
arXiv2022

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, λ}(\mathbb{B}^{n})>0$ such that $$S_{n, p, λ}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{p+1} \, {\rm d}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-λu^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\in C_c^{\infty}(\mathbb{B}^n),$ and $λ\leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-Δ_{\mathbb{B}^n}.$ It is known from the results of Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriate assumptions on $n,p$ and $λ$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,λ}(\mathbb{B}^n).$ In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in $\mathbb{R}^n$ of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.

math.AP↗
arXiv2022

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space: \begin{align}\label{abs:eqn} \left\{\begin{array}{ll} \partial_{t}u=Δ_{\mathbb{H}^{n}} u+ f(u, t) &\hbox{ in }~ \mathbb{H}^{n}\times (0, T),\\ \\ \quad u =u_{0} &\hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, it exhibits Fujita phenomena for the exponential weight $h(t) = e^{μt},$ i.e. there exists a critical exponent $μ^*$ such that if $μ> μ^*$ then all non-negative solutions blow-up in finite time and if $μ\leq μ^*$ there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that the above mentioned Cauchy problem with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

math.AP↗
arXiv2021

The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity

Let $Ω$ be a compact smooth domain containing zero in the Poincaré ball model of the Hyperbolic space $\mathbb{B}^{n}$ ($n \geq 3$) and let $-Δ_{\mathbb{B}^{n}}$ be the Laplace-Beltrami operator on $\mathbb{B}^{n}$, associated with the metric $g_{\mathbb{B}^{n}}= \frac{4}{(1-|x|^{2})^2}g_{_{\hbox{Eucl}}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray*} (E)~ \left\{ \begin{array}{lll} -Δ_{\mathbb{B}^{n}}u-γ{V_2}u -λu&=V_{2^\star(s)}|u|^{2^\star(s)-2}u &\hbox{ in }Ω\\ \hfill u &=0 & \hbox{ on } \partial Ω, \end{array} \right. \end{eqnarray*} where $0\leq γ\leq \frac{(n-2)^2}{4}$, $0< s <2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$ is the corresponding critical Sobolev exponent, $V_{2}$ (resp., $V_{2^\star(s)}$) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like $\frac{1}{r^{2}}$ (resp., $\frac{1}{r^{s}}$) at the origin. The bulk of this paper is a sharp blow-up analysis on approximate solutions of $(E)$ with bounded but arbitrary high energies. Our analysis leads to existence of positive ground state solutions for $(E)$, whenever $n \geq 4$, $0 \leq γ\leq \frac{(n-2)^2}{4}-1$ and $ λ> 0$. The latter result also holds true for $n\geq 3$ and $γ> \frac{(n-2)^2}{4}-1$ provided the domain has a positive "hyperbolic mass". On the other hand, the same analysis yields that if $γ> \frac{(n-2)^2}{4}-1$ and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to $(E)$, we show that there are infinitely many of them provided $n\geq 5$, $0\leq γ<\frac{(n-2)^2}{4}-4$ and $ λ> \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-γ\right)$.

math.AP↗
arXiv2020

Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial Ω$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(Ω)$ for the borderline Dirichlet problem, $-Δu-γ\frac{u}{|x|^2}- h(x) u = \frac{|u|^{{2^\star(s)}-2}u}{|x|^s}$ in $Ω$, where $0<s<2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$, $γ\in\mathbb{R}$ and $h\in C^0(\overlineΩ)$. We use sharp blow-up analysis on --possibly high energy-- solutions of corresponding subcritical problems to establish, for example, that if $γ<\frac{n^2}{4}-1$ and the principal curvatures of $\partialΩ$ at $0$ are non-positive but not all of them vanishing, then the above equation has an infinite number of (possibly sign-changing) solutions in ${H_{1,0}^2(Ω)}$. This complements results of the first and third authors, who had previously shown that if $γ\leq \frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partialΩ$ at $0$ is negative, then the equation has a positive solution. On the other hand, the sharp blow-up analysis also allows us to prove that if the mean curvature at $0$ is non-zero and if the mass (when defined) does not vanish, then there is a surprising stability under $C^1$-perturbations of the potential $h$ of those regimes where no variational positive solutions exist. In particular, and in sharp contrast with the non-singular case (i.e., when $γ=s=0$), we show non-existence of such solutions for (E) in any dimension, whenever $Ω$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.

math.AP↗
arXiv2018

Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space

We consider the Hardy-Schrödinger operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ on the Poincaré ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\sim \frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ is positive definite whenever $γ<\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-Δ_{\mathbb{B}^n}u-γ{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in }}\mathbb{B}^n,$$ where $0\leq s <2$, $2^*(s)=\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $Ω$ of ${\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < γ\leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. However, if $\frac{(n-2)^{2}}{4}-\frac{1}{4}< γ< \frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{_{\mathbb{B}^n}}(Ω)$ of the domain, a notion that we introduce and analyse therein.

math.AP↗
arXiv2017

Hardy-Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy

We study the asymptotic behavior of a sequence of positive solutions $(u_ε)_{ε>0}$ as $ε\to 0$ to the family of equations \begin{equation*} \left\{\begin{array}{ll} Δu_ε+a(x)u_ε= \frac{u_ε^{2^*(s_ε)-1}}{|x|^{s_ε}}& \hbox{ in }Ω\\ u_ε=0 & \hbox{ on }\partialΩ. \end{array}\right. \end{equation*} where $(s_ε)_{ε>0}$ is a sequence of positive real numbers such that $\lim \limits_{ε\rightarrow 0} s_ε=0$, $2^{*}(s_ε):= \frac{2(n-s_ε)}{n-2}$ and $Ω\subset \mathbb{R}^{n}$ is a bounded smooth domain such that $0 \in \partial Ω$. When the sequence $(u_ε)_{ε>0}$ is uniformly bounded in $L^{\infty}$, then upto a subsequence it converges strongly to a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain strong pointwise control on the blow up sequence, and then using the Pohozaev identity localize the point of singularity, which in this case can at most be one, and derive precise blow up rates. In particular when $n=3$ or $a\equiv 0$ then blow up can occur only at an interior point of $Ω$ or the point $0 \in \partial Ω$.

math.AP↗
arXiv2016

GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions

In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold $M$. We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist. To perform this topological argument, we overcome the difficulty of dealing with polyharmonic operators on a Riemannian manifold and adapting Lions's concentration-compactness lemma. Unlike Coron's original argument for a bounded domain in $\mathbb{R}^{n}$, we need to do more than chopping out a small ball from the manifold $M$. Indeed, our topological assumption that a small sphere on $M$ centred at a point $p \in M$ does not retract to a point in $M\backslash \{ p \}$ is necessary, as shown for the case of the canonical sphere where chopping out a small ball is not enough.

math.AP↗
arXiv2016

Struwe's Decomposition for a Polyharmonic Operator on a compact Riemannian Manifold with or without Boundary

Given a high-order elliptic operator on a compact manifold with or without boundary, we perform the decomposition of Palais-Smale sequences for a nonlinear problem as a sum of bubbles. This is a generalization of the celebrated 1984 result of Struwe. Unlike the case of second-order operators, bubbles close to the boundary might appear. Our result includes the case of a smooth bounded domain of $\mathbb{R}^n$.

math.AP↗