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Debabrata Karmakar

Publications and source records attributed to Debabrata Karmakar.

15 recordsLinked to original sources
arXiv2026

Gromov-Hausdorff Stability and Rigidity of manifolds via Heisenberg-Pauli-Weyl Uncertainty Principle

The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to Euclidean space, $\mathbb{R}^n$. This geometric discrepancy motivates the study of curvature dependent corrections and their associated stability properties. In this article, we investigate the geometric stability of the HPW inequality. Specifically, given a sequence of pointed Riemannian manifolds and appropriately normalized functions with a vanishing HPW deficit, we address whether the sequence converges to the corresponding model space in the pointed Gromov Hausdorff topology. We prove that for pinched Cartan Hadamard manifolds with sectional curvature bounded above by $c < 0$, the manifolds converge to the model hyperbolic space $\mathbb{H}^n_c$. In the non-negative Ricci curvature setting, we establish convergence to $\mathbb{R}^n$, up to a metric rescaling factor governed by the sequence's moment term. Consequently, we deduce that the known HPW inequality formulated via the asymptotic volume ratio (AVR) is suboptimal for non negatively Ricci curved manifolds not isometric to Euclidean space. To resolve this, we introduce a curvature corrected HPW inequality for this setting, analogous to the Cartan Hadamard case. Finally, we establish quantitative rigidity estimates in both curvature regimes, demonstrating that the HPW deficit when evaluated at Gaussian profiles which explicitly controls an appropriately defined distance to the respective model space.

math.DG↗
arXiv2026

A scalar field equation on hyperbolic space with indefinite sign nonlinearity

In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-Δ_{\mathbb{B}^N} u - λu = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$ on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $λ\in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases.

math.AP↗
arXiv2025

Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere

In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} α\int_{\mathbb{S}^2}|\nabla_{\mathbb{S}^2}u|^2 {\rm d}ω+2 \int_{\mathbb{S}^2} u {\rm d}ω-\frac{1}{2}\ln\left[\left(\int_{\mathbb{s}^2}e^{2u}{\rm d}ω\right)^2-\sum_{i=1}^3\left(\int_{\mathbb{s}^2}ω_i e^{2u}{\rm d} ω\right)^2\right] \geq 0 \end{align*} holds on $H^1(\mathbb{S}^2)$ if and only if $α\geq \frac{2}{3}$. In this regime, the infimum is attained only by trivial functions when $α> \frac{2}{3},$ whereas for the critical value $α= \frac{2}{3}$ nontrivial extremals exist, and Chang-Gui further provided a complete classification of such solutions. Building upon their result, we found a nice conformal invariance of the associated functional. Exploiting this invariance, we were able to characterize the full family of extremals in terms of conformal maps of $\mathbb{S}^2$ and, moreover, establish a sharp quantitative stability result in the gradient norm.

math.AP↗
arXiv2025

Existence of Bianchi-Egnell stability extremizer for the Hardy-Sobolev inequality

In this article, we prove the best Bianchi-Egnell constant for the Hardy-Sobolev (HS) inequality \begin{align*} C_{\tiny\mbox{BE}}(γ) := \inf_{u \ \small \mbox{not an optimizer}} \frac{\int_{\mathbb{R}^n} \left(|\nabla u|^2 - \fracγ{|x|^2}u^2\right) \ {\rm d}x - S_γ\|u\|_{L^{2^{\star}}}^2}{\mbox{dist} (u, \ \mbox{set of optimizers})^2}, \end{align*} is attained, extending the result of König [arXiv:2211.14185] for the classical Sobolev inequality (that corresponds to $γ= 0$). One of the main difficulties is that the third eigenspace of the linearized operator may contain only spherical harmonics of degree $1$, and hence, an essential non-vanishing criterion fails [arXiv:2210.08482]. This non-vanishing criterion is indispensable for proving the best Bianchi-Egnell constant $C_{\tiny\mbox{BE}}(γ) < C_{\tiny\mbox{BE}}^{\tiny\mbox{loc}}(γ)$ that prevents a minimizing sequence converging to one of the optimizers. In addition, not being translation invariant, extracting a non-zero weak limit from a minimizing sequence presents difficulties. We found another hidden critical level $C_{\tiny\mbox{BE}}(γ) <1 - \frac{S_γ}{S},$ where $S$ is the best Sobolev constant that plays a significant role in proving the existence of an extremizer. In particular, we show that there exists a $γ_0>0$ such that for $γ\geq γ_0,\ C_{\tiny\mbox{BE}}(γ)$ is attained. Moreover, we remark that there is a region $γ_0 \leq γ< γ_c^{\star},$ where the third eigenspace of the linearized operator contains only spherical harmonics of degree $1.$ Our result improves some of the results in Wei-Wu [arXiv:2308.04667] corresponding to the HS inequality.

math.AP↗
arXiv2025

Total masses of solutions to general Toda systems with singular sources

In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compact surfaces. The total mass is found to be related to the longest element $κ$ in the Weyl group of the corresponding Lie algebra. This is the foundation to future work relating the local blowup masses (from analysis) with the Weyl group. This work generalizes the previous works in Lin et al. (2012), Ao et al. (2015) and Nie (20160 for Toda systems of types $A, G_2$ and $B, C$. However, a more Lie-theoretic method is needed here for the general case, and the method relies heavily on the DPW method, Drinfeld-Sokolov gauge and the $W$-invariants. The last crucial step for the total masses is obtained by applying the work of Kostant (1979) on the one dimensional Toda lattice.

math.AP↗
arXiv2025

A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space \begin{align*} Δ_{\mathbb{B}^N}u+λu +|u|^{p-1}u+θu \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N, \end{align*} and study the existence vs non-existence results. We show that whenever $θ>0,$ there exists an $H^1$-solution, while for $θ<0$, there does not exist a positive solution in a reasonably general class. Since the perturbation $ u \ln u^2$ changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an "almost" precise lower asymptotic decay estimate on the positive solutions for $θ<0,$ culminating in proving their non-existence assertion.

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

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↗
arXiv2020

On the critical mass Patlak-Keller-Segel system for multi-species populations: global existence and infinite time aggregation

We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline case of critical mass there exists a global {\it free energy solution} subject to initial data with finite entropy and second moment. Moreover, we show that as time $t$ approaches to infinity, all the components of the solutions concentrate in the form of a Dirac measure at a single point. Our approach utilizes the gradient flow structure in Wasserstein space in the spirit of De Giorgi's minimizing movement or the JKO-schemes. Due to the critical mass, the minimization problem in JKO-schemes may not admit a solution in general. We find a necessary and sufficient criterion for which any minimizing sequence remains uniformly bounded in an appropriate topology to ensure the existence of a minimizer.

math.AP↗
arXiv2019

On Patlak-Keller-Segel system for several populations: a gradient flow approach

We study the global in time existence of solutions to the parabolic-elliptic Patlak-Keller-Segel system of multi-species populations. We prove that if the initial mass satisfies an appropriate notion of sub-criticality, then the system has a solution defined for all time. We explore the gradient flow structure in the Wasserstein space to study the question of existence. Moreover, we show that the obtained solution satisfies energy dissipation inequality.

math.AP↗
arXiv2018

Self-similar solutions of decaying Keller-Segel systems for several populations

It is known that solutions of the parabolic elliptic Keller-Segel equations in the two dimensional plane decay, as time goes to infinity, provided the initial data admits sub-critical mass and finite second moments, while such solution concentrate, as $t\rightarrow\infty$, in the critical mass. In the sub-critical case this decay can be resolved by a steady, self-similar solution, while no such self similar solution is known to exist for the concentration in the critical case. This paper is motivated by the Keller-Segel system of several interacting populations, under the existence of an additional drift for each component which decays in time at the rate $O(1/\sqrt{t})$. We show that self-similar solutions always exists in the sub-critical case, while the existence of such self-similar solution in the critical case depends on the gap between the decaying drifts for each of the components. For this, we study the conditions for existence/non existence of solutions for the corresponding Liouville's systems, which, in turn, is related to the existence/non existence of minimizers to a corresponding Free Energy functional.

math.AP↗
arXiv2015

Adams' inequality with exact growth in the hyperbolic space $\mathbb{H}^4$ and Lions lemma

In this article we prove Adams inequality with exact growth condition in the four dimensional hyperbolic space $\mathbb{H}^4,$ \begin{align} \int_{\mathbb{H}^4} \frac{e^{32 π^2 u^2} - 1}{(1 + |u|)^2} \ dv_g \leq C ||u||^2_{L^2({\mathbb{H}^4})}. \end{align} for all $u \in C^{\infty}_c(\mathbb{H}^4) $ with $\int_{\mathbb{H}^4} (P_2 u) u \ dv_g \leq 1.$ We will also establish an Adachi-Tanaka type inequality in this settings. Another aspect of this article is the P.L.Lions lemma in the hyperbolic space. We prove P.L.Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

math.AP↗
arXiv2015

Adam's Inequality on the Hyperbolic space

In this article we establish an Adam's Inequality in the Hyperbolic space. As an application we will also prove the asymptotic behaviour of the best constants in the Sobolev inequality and also discuss the solvability of Q curvature type PDE's in the Hyperbolic space.

math.AP↗