Related papers: Frequency theorem for parabolic equations and its …

In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted…

Lyapunov functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify…

A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace…

We derive spectral fluctuation--dissipation--response inequalities for finite-state Markov jump processes. By comparing the causal susceptibility to its passive equilibrium reference, we establish frequency-resolved and frequency-integrated…

We consider a frequency localized Bernstein inequality for the fractional Laplacian operator which has wide applications in fluid dynamics such as dissipative surface quasi-geostrophic equations. We use a heat flow reformulation and prove…

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture,…

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfu{\ss} conjecture…

Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several…

In this paper, we study the frequency response of uncertain systems using Kharitonov stability theory on first order complex polynomial set. For an interval transfer function, we show that the minimal real part of the frequency response at…

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…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary…

Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a compact Lie group $G$, and $P_0$ an invariant elliptic classical pseudodifferential operator on $M$. Using Fourier integral operator…

A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on…

We develop a unified fluctuation-response theory in the frequency domain for nonequilibrium steady states governed by overdamped Langevin dynamics and Markov jump processes. The relation expresses the power spectrum of general observables…

This paper studies the uniformly asymptotic stability of nonautonomous systems on Riemannian manifolds. We establish corresponding Lyapunov-type theorems (Theorems 2.1 and 2.2), extending classical Euclidean results (e.g., [9, Theorems 4.9…

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum…

Within the method of spectral moments it is possible to construct the spectral function of a many-electron system from the first $2P$ spectral moments ($P=1,2,3,\dots$). The case $P=1$ corresponds to standard Kohn-Sham density functional…

For systems evolving on a Riemannian manifold, we propose converse Lyapunov theorems for asymptotic and exponential stability. The novelty of the proposed approach is that is does not rely on local Euclidean coordinate, and is thus valid on…

For strongly continous semigroups on Hilbert spaces, we investigate admissibility properties of control and observation operators shifted along continuous scales of spaces built by means of either interpolation and extrapolation or…