Liouville Type Property Research Articles

Liouville-type results for fully nonlinear subelliptic equations on the Heisenberg group

ABSTRACT In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations.

Uniform Structural Stability of Hagen–Poiseuille Flows in a Pipe

In this paper, we prove the uniform nonlinear structural stability of Hagen–Poiseuille flows with arbitrary fluxes in the axisymmetric case in an infinitely long pipe. This uniform nonlinear structural stability is the first step to study Liouville type property for steady solutions of Navier–Stokes system in a pipe, which may play an important role in proving the existence of solutions with arbitrary flux to steady Navier–Stokes system in a nozzle with Poiseuille flows as far field asymptotic states (Leray’s problem). A key step is the a priori estimate for the associated linearized problem for Navier–Stokes system around Hagen–Poiseuille flows. The linear structural stability is established as a consequence of elaborate analysis on the governing equation for the partial Fourier transform of the stream function. The uniform estimates are obtained based on the analysis for the solutions with different fluxes and frequencies. One of the most involved cases is to analyze the solutions with large flux and intermediate frequency, where the boundary layer analysis plays a crucial role.

Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds

This paper aims at deriving apriori bounds on the gradient of positve solutions to a class of semilinear elliptic equation, with applications focusing on establishing a liouville type property for the bounded solutions of the Allen-Cahn equation on a complete noncompact Riemannian manifold with nonnegative Ricci curvature.

Harnack inequalities for a class of heat flows with nonlinear reaction terms

A class of semilinear heat flows with general nonlinear reaction terms is considered on complete Riemannian manifolds with Ricci curvature bounded from below. Two types of (space-time and space only) gradient estimates are established for positive solutions to the flow, and the corresponding Harnack inequalities are obtained to allow for comparison of solutions. Some specific examples of the reaction term such as logarithmic reaction, Fisher-KPP and Allen-Cahn equations are discussed as applications of the estimates so derived. Referring to logarithmic nonlinearities, some discussions are made on Liouville type properties of bounded solutions.

On a two dimensional nonlocal shallow-water model

In the present study we describe the asymptotic perturbation method to derive a two-dimensional nonlocal shallow-water model equation in the context of full water waves. Starting from the incompressible and irrotational governing equations in the three-dimensional water waves, we show that such a equation arises in the modeling of the propagation of shallow water waves over a flat bed. The resulting equation is a two dimensional Camassa-Holm equation-type with weakly transverse effect for the horizontal velocity component. The equation captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg-de Vries and Kadomtsev-Petviashvili equations. We also address some properties of this model equation and how it relates to the surface wave. Analytically, we establish the local well-posedness of this model in a suitable Sobolev space. We then investigate formation of singularities and existence of traveling-wave solutions to this quasi-linear model equation with an emphasis on the understanding of weak transverse effect. Finally, we provide a Liouville-type property to obtain unique continuation result for the strong solution.

Orbital stability of solitary waves and a Liouville-type property to the cubic Camassa–Holm-type equation

We classify the existence and non-existence of a class of localized solitary waves for the cubic Camassa–Holm-type equation which arises as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves. In addition to those peaked solitary-wave solutions, we show by the phase portrait method that there are smooth and cusped solitary waves according to the wave speed and coefficients of nonlocal linear and nonlinear dispersions. We then prove that the smooth solitary-wave solutions are orbitally stable in the energy space under small perturbation. Finally, we establish a Liouville-type property of the solution.

Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form \({\mathbb {R}}^N\setminus K\), where \(K\subset {\mathbb {R}}^N\) is a compact “obstacle”. The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as \(t\rightarrow -\infty \) and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.

Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

Abstract We consider the elliptic quasilinear equation - Δ m ⁢ u = u p ⁢ | ∇ ⁡ u | q {-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝ N {\mathbb{R}^{N}} , q ≥ m {q\geq m} and p > 0 {p>0} , 1 < m < N {1<m<N} . Our main result is a Liouville-type property, namely, all the positive C 1 {C^{1}} solutions in ℝ N {\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝ N ∖ B r 0 {\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in B r 0 ∖ { 0 } {B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in B r 0 {B_{r_{0}}} . The solutions in ℝ N ∖ { 0 } {\mathbb{R}^{N}\setminus\{0\}} has a finite limit l ≥ 0 {l\geq 0} as | x | → ∞ {\lvert x\rvert\to\infty} . Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

We prove new (sharp) Liouville-type properties via degenerate Hadamard three-sphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci’s extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework.

Liouville-type theorems for the stationary MHD equations in 2D

This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some Lp norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.

On Defining Functions and Cores for Unbounded Domains II

We continue the study of the core $${\mathfrak {c}}(\Omega )$$ of a complex manifold $$\Omega $$ which was initiated in Harz et al. (Math Z, doi:10.1007/s00209-016-1792-9, 2016). The main focus lies on the investigation of pseudoconcavity and Liouville-type properties of the core. We construct examples of strictly pseudoconvex domains $$\Omega \subset {\mathbb {C}}^n$$ with nonempty core which show that, first, $${\mathfrak {c}}(\Omega )$$ may not contain any analytic variety of positive dimension, and, second, it can happen that $${\mathfrak {c}}(\Omega )$$ is connected but not every smooth and bounded from above plurisubharmonic function on $$\Omega $$ is constant on $${\mathfrak {c}}(\Omega )$$. However, we show that for each complex manifold of dimension two, there exists a decomposition of the core into disjoint connected sets which are 1-pseudoconcave in $$\Omega $$ and satisfy a Liouville-type theorem with respect to smooth plurisubharmonic functions on $$\Omega $$. We also introduce cores of higher order, by requiring not only failure of strict plurisubharmonicity of smooth and bounded from above plurisubharmonic functions $$\varphi $$ on $$\Omega $$, but also instead by prescribing an upper bound k for the rank of the complex Hessian of $$\varphi $$, with k possibly different from $$\dim _{\mathbb {C}}\Omega -1$$. We show that this sharpening of the definition does not lead to better results on pseudoconcavity of the core.

A Note on Harmonic Functions on Surfaces

We review and give elementary proofs of Liouville-type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature.

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

Qualitative properties of solutions for an integral system related to the Hardy–Sobolev inequality

This article carries out a qualitative analysis on a system of integral equations of the Hardy–Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are established. For a bounded and decaying positive solution, it is shown that it either decays with the slow rates or the fast rates depending on its integrability. Particularly, a criterion for distinguishing integrable solutions from other bounded and decaying solutions in terms of their asymptotic behavior is provided. Moreover, related results on the optimal integrability, boundedness, radial symmetry and monotonicity of positive integrable solutions are also established. As a result of the equivalence between the integral system and a system of poly-harmonic equations under appropriate conditions, the results translate over to the corresponding poly-harmonic system. Hence, several classical results on semilinear elliptic systems are recovered and further generalized.

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$ u_t(t,x)=(\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in\mathcal{H}, $$ where $\mathcal{H}=\RR^N$ or $\ZZ^N$, $\mathcal{A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal{H}=\RR^N$ and is a discrete dispersal operator in the case $\mathcal{H}=\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\|x\|\to\infty$ for some time and space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.

Liouville-Type Theorems for the Forced Euler Equations and the Navier–Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |\Delta v|^{\frac65} dx<\infty$, which is stronger than the important D-condition, $\int_{\Bbb R^3} |\nabla v|^2 dx <\infty$, but both having the same scaling property, implies that $v=0$. In the appendix we reprove the Theorem 1.1(\cite{cha0}), using the self-similar Euler equations directly.

POSITIVE p-HARMONIC FUNCTIONS ON GRAPHS

Suppose that an inflnite graph G of bounded degree has flnite number of ends, each of which is p-regular, where 1 < p < 1. Then we can identify all the positive (bounded, respectively) p-harmonic functions on G. In this paper, we study the Liouville type property on a graph. By the Liouville property on a graph G, we mean that every bounded har- monic function on G is constant. It immediately follows that the set of all bounded harmonic functions on G having Liouville property is in one to one correspondence with the real line R. It seems natural to regard the case that the set of all bounded harmonic functions on G is in one to one correspondence with R l for some positive integer l as a general- ized version of the Liouville property. In line with this view point, we consider the case of the p-Laplacian operator (1 < p < 1) and the posi- tive (bounded, respectively) p-harmonic functions on graphs of bounded degree. In the case that a graph G has a flnite number of ends, each of which is p-regular, we identify all the positive (bounded, respectively) p-harmonic functions on G in section 4. On the other hand, Kanai(5) proved that if G and H are roughly isometric graphs of bounded degree, then H is parabolic whenever G is. In (9), Soardi proved that if G and H are roughly isometric graphs of bounded degree, and if G has no nonconstant harmonic functions with flnite energy, then neither has H. Later, the second author (8) proved that the dimension of the space of harmonic functions with flnite energy is preserved under rough isometries between graphs of bounded degree. In this paper, we also discuss its rough isometric invariance in section 3

A Liouville property and quasiconvergence for a semilinear heat equation

This paper is concerned with a supercritical semilinear diffusion equation with the power nonlinearity. Via establishing a Liouville-type property, we prove the quasiconvergence (convergence to a set of steady states) of a large class of global solutions. The method of proof relies on similarity variables and invariant manifold ideas.

Hadamard and Liouville Type Results for Fully Nonlinear Partial Differential Inequalities

In this paper we prove some Hadamard and Liouville type properties for nonnegative viscosity supersolutions of fully nonlinear uniformly elliptic partial differential inequalities in the whole space.

Removable singularities and Liouville-type property of analytic multivalued functions

Removable singularities and Liouville-type property of analytic multivalued functions