Liouville Theorem Research Articles

Liouville-type theorem for higher order Hardy-Hénon type systems on the sphere

In this paper, we study Liouville type theorems for the positive solutions to the following higher order Hardy-Hénon type system involving the conformal GJMS operator on the sphere Sn. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere Sn into a higher order elliptic system on Rn. Then, we show that every positive solution of the higher order elliptic system on Rn is a solution to the associated integral system on Rn by using polyharmonic average and iteration arguments. We use the method of moving planes in integral form to prove that there are no positive solutions for the integral system on Rn. Finally, together with the symmetry of the sphere Sn, we obtain the Liouville type theorem of the higher order Hardy-Hénon type system involving the GJMS operator on the sphere. The results of this paper are also new even for the Lane-Emden system on the sphere.

Remarks on the Anisotropic Liouville Theorem for the Stationary Tropical Climate Model

Remarks on the Anisotropic Liouville Theorem for the Stationary Tropical Climate Model

Liouville type theorems for subelliptic systems on the Heisenberg group with general nonlinearity

In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group \(\mathbb{H}^{n}\) involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane–Emden system on the Heisenberg group \(\mathbb{H}^{n}\), which also appears to be a new result in the literature.

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Temporal Evolution of Electron Accelerations behind the Dipolarization Front in the Earth's Magnetotail

We report the temporal evolution of electron pitch angle distributions behind the dipolarization front (DF) in the Earth's magnetotail with observations of the Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft. Taking advantage of multipoint observations from the THEMIS mission lined up in space, we study pitch angle distributions of energetic electrons behind the DF during two typical events. Pancake, rolling-pin, and cigar distributions are observed sequentially during the acceleration process. Based on Liouville's theorem, it is revealed that pancake distribution is dominantly formed by betatron acceleration in the early stage, and rolling-pin distribution is generated by both dominant Fermi and weak betatron acceleration in the transition stage, while cigar distribution is formed by Fermi acceleration finally. Our results provide comprehensive in situ observational evidence of the temporal evolution of electron acceleration behind the DF during propagation.

Liouville Theorems for a Class of Degenerate or Singular Monge–Ampère Equations

Liouville Theorems for a Class of Degenerate or Singular Monge–Ampère Equations

A view on Liouville theorems in PDEs

Abstract Our review of Liouville theorems includes a special focus on nonlinear partial differential equations and inequalities.

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

AbstractIn this article we study local and global properties of positive solutions of − Δmu = ∣u∣p−1u+M∣∇u∣q in a domain Ω of ℝN, with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.

Analysis of harmonic functions under lower bounds of N-weighted Ricci curvature with ε-range

The behavior of harmonic functions on Riemannian manifolds under lower bounds of the Ricci curvature has been studied from both analytic and geometric viewpoints. For example, some Liouville type theorems are obtained under lower bounds of the Ricci curvature. Recently, those results are generalized under lower bounds of the N-weighted Ricci curvature RicψN with N∈[n,∞]. In this paper, we present a Liouville type theorem for harmonic functions of sublinear growth and a gradient estimate of harmonic functions on weighted Riemannian manifolds under weaker lower bounds of RicψN with N<0. We also prove an Lp-Liouville theorem under lower bounds of RicψN with N∈[n,∞] in a different way from previous studies. Our results are obtained as a consequence of an argument under lower bounds of RicψN with ε-range, which is a unification of constant and variable curvature bounds. Among various methods considered for the analysis of harmonic functions, this paper focuses on methods using the Moser's iteration procedure.

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .

Rigidity of the Delaunay triangulations of the plane

We prove a rigidity result for Delaunay triangulations of the plane under Luo's notion of discrete conformality, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We follow Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and develop a discrete Schwarz lemma and a discrete Liouville theorem. As a key ingredient to prove the discrete Schwarz lemma, we establish a correspondence between the Euclidean discrete conformality and the hyperbolic discrete conformality, for geodesic embeddings of triangulations. Other major tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry.

Schauder estimates for parabolic equations with degenerate or singular weights

We establish some C0,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{0,\\alpha }$$\\end{document} and C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} as a power a>-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a > -1$$\\end{document} of the distance to Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

Radial symmetry and Liouville theorem for master equations

Radial symmetry and Liouville theorem for master equations

Liouville theorems for a fourth order Hénon equation in the half-space

We investigate here the nonlinear elliptic Hénon-type equation \Delta^{2}u= |x|^{a}|u|^{p-1}u \quad \text{in }\mathbb{R}^{n}_{+}, \quad u =\frac{\partial u}{\partial x_{n}}= 0 \quad \text{in }\partial \mathbb{R}^{n}_{+}, where p&gt;1 , a\geq 0 and n\geq 5 . Based on the approach of Hu [J. Differential Equations 256 (2014), 1817–1846], we prove Liouville-type theorems for stable solutions and solutions which are stable outside a compact set possibly unbounded and sign-changing. In contrast with the results of Hu (2014), we apply a new method to provide an implicit existence of the fourth-order Joseph–Lundgren exponent. To classify finite Morse index solutions in the supercritical case, we adopt a new method of monotonicity formula together with blowing down sequence. In addition, a difficulty stems from the fact that applying the doubling lemma leads to the singularity. For this reason, we use a more delicate approach to the interval (n+4+2a, p_{JL2}(n,0)) . Our analysis uses a combination of some integral estimates, Pohozaev-type identity, and monotonicity formula of solutions.

A Liouville theorem for elliptic equations with a potential on infinite graphs

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u≡0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\equiv 0$$\\end{document}. We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

Machian fractons, Hamiltonian attractors, and nonequilibrium steady states

We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N, despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of “order-by-disorder” borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics. Published by the American Physical Society 2024

Mathematical analysis of acoustic wave interaction of vibrating rigid plates with subsonic flows

This research article investigates the effects of object's vibration and fluid movement on the acoustics of subsonic flows, specifically focusing on the scattering of acoustic waves by a vibrating rigid plate submerged in uniform flow. The acoustic plane wave that impacts the plate, coupled with its oscillatory motion, causes a disruption in the fluid medium. This disturbance, in turn, gives rise to a Rayleigh wave that propagates along the boundary separating the plate and the fluid. The study uses the Wiener‐Hopf technique to analytically model the acoustic scattering by a rigid barrier of finite dimensions and analyze the relationship between acoustics and structures. The method involves applying Fourier transformations to the governing boundary value problem and resolving the Wiener‐Hopf equations using the factorization theorem, Liouville's theorem, and analytical continuation. The integral equations of scattered potential computed asymptotically are used to describe the acoustic characteristics of structures and their interaction with fluid flow in subsonic conditions. The findings of the study reveal the sharp peaks of the scattered potential at certain angles with more oscillation in high subsonic flow. Also, increasing the frequency of the vibrating plate increases the amplitude of the scattered potential but is attenuated in mean flow whereas enhancing plate vibrations amplifies the scattered sound, and it is more vibrant in high subsonic flow than mean flow and no fluid flow. This research has applications in noise reduction, aeronautical engineering, and the detection of underwater structures using acoustic waves and micropolar elastic media.

An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

We study the equation m(D)f=0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m \in C(\mathbb{R}^{n}) , we show that every such solution can be analytically continued to a sub-exponentially growing entire function on \mathbb{C}^{n} if, and only if, m(\xi) \neq 0 for \xi \neq 0 .

A Liouville theorem for F-harmonic maps with moderate divergent energy

In this paper, we obtain a Liouville theorem of F-harmonic maps between complete Riemannian manifolds with moderate divergent F-energy. We assume that F is a concave function and satisfies a differential inequality. We employ Ara's F-stress-energy tensor and the Hessian comparison theorem to prove the main result.