arXiv subjects

Mayukh Mukherjee

Publications and source records attributed to Mayukh Mukherjee.

28 records · Page 2Linked to original sources

arXiv2020

Polyhedral billiards, eigenfunction concentration and almost periodic control

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.

math.AP↗

arXiv2017

Nodal Geometry, Heat Diffusion and Brownian Motion

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.

math.DG↗

arXiv2017

Some remarks on nodal geometry in the smooth setting

We consider a Laplace eigenfunction $φ_λ$ on a smooth closed Riemannian manifold, that is, satisfying $-Δφ_λ= λφ_λ$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $φ_λ$. This extends previous work of Jakobson and Mangoubi in case $ (M, g) $ is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. \cite{L1}). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $Ω_λ$ on which $φ_λ$ satisfies exponentially small $L^\infty$ bounds, we give some quantitative estimates for radii of inscribed balls.

math.AP↗

arXiv2017

On maximizing the fundamental frequency of the complement of an obstacle

Let $Ω\subset \mathbb{R}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let $ D $ be an arbitrary bounded domain referred to as "obstacle". We are interested in the behaviour of the first Dirichlet eigenvalue $ λ_1(Ω\setminus (x+D)) $. First, we prove an upper bound on $ λ_1(Ω\setminus (x+D)) $ in terms of the distance of the set $ x+D $ to the set of maximum points $ x_0 $ of the first Dirichlet ground state $ ϕ_{λ_1} > 0 $ of $ Ω$. In short, a direct corollary is that if \begin{equation} μ_Ω:= \max_{x}λ_1(Ω\setminus (x+D)) \end{equation} is large enough in terms of $ λ_1(Ω) $, then all maximizer sets $ x+D $ of $ μ_Ω$ are close to each maximum point $ x_0 $ of $ ϕ_{λ_1} $. Second, we discuss the distribution of $ ϕ_{λ_1(Ω)} $ and the possibility to inscribe wavelength balls at a given point in $ Ω$. Finally, we specify our observations to convex obstacles $ D $ and show that if $ μ_Ω$ is sufficiently large with respect to $ λ_1(Ω) $, then all maximizers $ x+D $ of $ μ_Ω$ contain all maximum points $ x_0 $ of $ ϕ_{λ_1(Ω)} $.

math.AP↗

arXiv2017

Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary condition. Classically known results, and also recent work in \cite{DOS} and \cite{DM} establish sufficient conditions for $L^\infty-\text{BMO}_L$ continuity of $φ(\sqrt{A})$, where $φ\in S^0_1(\mathbb{R})$, and $A$ is a suitable elliptic operator. Using a variant of the Cheeger-Gromov-Taylor functional calculus due to \cite{MMV}, and short time bounds on the integral kernel of $e^{tL}$ due to \cite{G}, we prove that a variant of such sufficient conditions holds for our operator $L$.

math.AP↗

arXiv2017

A Spectral Gap Estimate and Applications

We consider the Schrödinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $λ_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\text{ where } I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$ λ_1 \geq \frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y \right\}$ is an interval for the value of $y$ solving the minimization problem. An immediate application is as follows: let $Ω\subset \mathbb{R}^2$ be a convex domain with inradius $ρ$ and diameter $D$ and let $u:Ω\rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-Δ$ on $Ω$ with Dirichlet boundary conditions on $\partial Ω$. We prove $$ \| u \|_{L^{\infty}} \lesssim \frac{1}{ρ^{}} \left( \fracρ{D} \right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions.

math.SP↗

arXiv2016

Discrete $z$-filters and rings of analytic functions

Consider rings of single variable real analytic or complex entire functions, denoted by $\mathbb{K}\langle z\rangle$. We study "discrete $z$-filters" on $\mathbb{K}$ and their connections with the space of maximal ideals of $\mathbb{K}\langle z\rangle$, which we characterize as a compact $T_1$ space $θ\mathbb{K}$ of discrete $z$-ultrafilters on $\mathbb{K}$. We show that $θ\mathbb{K}$ is a bijective continuous image of $β\mathbb{K} \setminus Q(\mathbb{K})$, where $Q(\mathbb{K})$ is the set of far points of $β\mathbb{K}$. $θ\mathbb{K}$ turns out to be the Wallman compactification of the canonically embedded image of $\mathbb{K}$ inside $θ\mathbb{K}$. Using our characterization of $θ\mathbb{K}$, we derive a Gelfand-Kolmogorov characterization of maximal ideals of $\mathbb{K}\langle z\rangle$ and show that the Krull dimension of $\mathbb{K}\langle z\rangle$ is at least $c$. We also establish the existence of a chain of prime $z$-filters on $\mathbb{K}$ consisting of at least $2^c$ many elements.

math.GN↗

arXiv2016

Variation of Laplace spectra of compact "nearly" hyperbolic surfaces

We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in ~\cite{B}, namely, we prove that the Laplace spectra of negatively curved compact surfaces having same genus $γ\geq 2$, same area and same curvature bounds vary in a "controlled way", of which we give a quantitative estimate (Theorem 1.1 below). We also observe how said real analyticity can lead to unexpected conclusions about spectral properties of generic metrics on a compact surface of genus $γ\geq 2$ (Proposition 1.5 below).

math.DG↗

arXiv2015

Nonlinear travelling waves on non-Euclidean spaces

We study travelling wave solutions, that is, solutions of the form $v(t, x) = e^{iλt}u(g(t)x)$, to nonlinear Schrödinger and Klein-Gordon equations on Riemannian manifolds, both compact and non-compact ones, with emphasis on the NLKG. Here $g(t)$ represents a one-parameter family of isometries generated by a Killing field $X$ and a case of particular interest is when $X$ has length $\leq 1$, which leads in certain settings to hypoelliptic operators with loss of at least one derivative. In the compact case, we establish existence of travelling wave solutions via "energy" minimization methods and prove that at least compact isotropic manifolds have \emph{genuinely} travelling waves. We establish certain sharp regularity estimates on low dimensional spheres that improve results in ~\cite{T1} and carry out the subelliptic analysis for NLKG on spheres of higher dimensions utilizing their homogeneous coset space properties. Such subelliptic phenomenon have no parallel in the setting of flat spaces. We will also study related phenomenon on complete noncompact manifolds with certain symmetry assumptions using concentration-compactness type arguments.

math.AP↗

arXiv2015

Extremal values of the (fractional) Weinstein functional on the hyperbolic space

We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space $\mathbb{H}^n$. We are primarily interested in the existence of Weinstein functional maximisers, or, in other words, existence of extremal functions for the best constant of the Gagliardo-Nirenberg inequality. The main result is that the maximum value of the Weinstein functional on $\mathbb{H}^n$ is the same as that on $\mathbb{R}^n$ and the related fact that the maximum value of the Weinstein functional is not attained on $\mathbb{H}^n$, when maximisation is done in the Sobolev space $H^1(\mathbb{H}^n)$. This proves a conjecture made in ~\cite{CMMT} and also answers questions raised in several other papers (see, for example, ~\cite{B}). We also prove that a corresponding version of the conjecture will hold for the Weinstein functional with the fractional Laplacian as well.

math.AP↗