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Debdip Ganguly

Publications and source records attributed to Debdip Ganguly.

At least 37 records · Page 2Linked to original sources
arXiv2022

The Fractional Porous Medium Equation on the hyperbolic space

We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $Δ_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples.

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

Fractional Hardy equations with critical and supercritical exponents

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-Δ)^s u -θ\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\, \mathbb{R}^N, \quad u \in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), \end{align*} where $s\in (0,1)$, $N>2s$, $q>p\geq{(N+2s)}/{(N-2s)}$, $θ\in(0, Λ_{N,s})$ and $Λ_{N,s}$ is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions we mean, both the radial symmetry that is obtained by using the moving plane method in a nonlocal setting on the whole $\mathbb{R}^N$, and upper bound behavior of the solutions. To this last end we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

math.AP↗
arXiv2021

Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs

We prove a family of Hardy-Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.

math.AP↗
arXiv2020

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality cannot be further improved. Such inequalities arise from more general, \emph{optimal} ones valid for the operator $ P_λ:= -Δ_{\mathbb{H}^N} - λ$ where $0 \leq λ\leq λ_{1}(\mathbb{H}^N)$ and $λ_{1}(\mathbb{H}^N)$ is the bottom of the $L^2$ spectrum of $-Δ_{\mathbb{H}^N} $, a problem that had been studied in [J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the operator $P_{λ_{1}(\mathbb{H}^N)}$. A different, critical and new inequality on $\mathbb{H}^N$, locally of Hardy type, is also shown. Such results have in fact greater generality since there are shown on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_λ.$

math.AP↗
arXiv2020

On some strong Poincaré inequalities on Riemannian models and their improvements

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.

math.FA↗
arXiv2020

On the first frequency of reinforced partially hinged plates

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eingenvalue problem. In order to improve the stability of the plate, it seems reasonable to place a certain amount of stiffening material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In the present paper we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

math.AP↗
arXiv2020

Improved L$^p$-Poincaré inequalities on the hyperbolic space

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré-Hardy inequality, namely an improvement of the best $p$-Poincaré inequality in terms of the Hardy weight $r^{-p}$, $r$ being geodesic distance from a given pole. Certain Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.

math.FA↗
arXiv2019

Existence and Multiplicity of positive solutions of certain nonlocal scalar field equations

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left\{\begin{aligned} (-Δ)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, u &\in H^{s}{(\mathbb{R}^{N})} \end{aligned} \right. \end{equation} where $s \in (0,1)$, $N>2s$, $1<p<2_s^*-1:=\frac{N+2s}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^N)$ and $f\in H^{-s}(\mathbb{R}^N)$ is a nonnegative functional i.e., ${\langle}f,u{\rangle} \geq 0$ whenever $u$ is a nonnegative function in $H^s(\mathbb{R}^N)$. We prove existence of a positive solution when $f\equiv 0$ under certain asymptotic behavior on the function $a.$ Moreover, when $a(x)\geq 1$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$), then we show that the above equation admits at least two positive solutions. Finally, we establish existence of three positive solutions to the above equation, under the condition that $a(x)\leq 1$ with $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{H^{-s}(\mathbb{R}^N)}$ is small enough (but $f\not\equiv 0$).

math.AP↗
arXiv2019

Integral representation using Green function for fractional Hardy equation

Our main aim is to study Green function for the fractional Hardy operator $P:=(-Δ)^s -\fracθ{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<θ<Λ_{N,s}$ and $Λ_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

math.AP↗
arXiv2018

Improved Multipolar Poincaré-Hardy inequalities on Cartan-Hadamard Manifolds

We prove a family of improved multipolar Poincaré-Hardy inequalities on Cartan-Hadamard manifolds. For suitable configurations of poles, these inequalities yield an improved multipolar Hardy inequality and an improved multipolar Poincaré inequality such that the critical unipolar singular mass is reached at any pole.

math.AP↗
arXiv2018

Certain Liouville properties of eigenfunctions of elliptic operators

We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.

math.FA↗
arXiv2017

On the Equivalence of Heat Kernels of Second-order parabolic operators

Let $P$ be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold $M$, and let $V$ be a real valued function which belongs to the class of {\em small perturbation potentials} with respect to the heat kernel of $P$ in $M$. We prove that under some further assumptions (satisfying by a large classes of $P$ and $M$) the positive minimal heat kernels of $P-V$ and of $P$ on $M$ are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic

math.AP↗
arXiv2017

On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu = 0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive solution, then such a solution is called a ground state of P in M, and P is said to be critical in M. We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P^{\star}. Under some further assumptions, we describe the behaviour at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace-Beltrami operator on a smooth and complete Riemannian manifold M.

math.AP↗
arXiv2016

Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-Δ_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-Δ_{\mathbb H^N}$. We find the optimal constant in the resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at least one of the constant involved is again sharp.

math.CA↗
arXiv2015

Improved higher order Poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.

math.CA↗