Related papers: Liouville's theorem for the generalized harmonic f…
In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and…
In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes…
The classical Fatou theorem identifies bounded harmonic functions on the unit disk with bounded measurable functions on the boundary circle. We extend this theorem to bounded harmonic maps.
We prove that a Hamilton-Jacobi equation in 1D with periodic forcing has a set of generalized solutions such that each solution is a sum of linear and continuous periodic functions; we also give a condition of uniqueness of this solution in…
We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their…
We study certain typical semilinear elliptic equations in Euclidean space $\bR^{n}$ or on a closed manifold $M$ with nonnegative Ricci curvature. Our proof is based on a crucial integral identity constructed by the invariant tensor method.…
The mean flux theorems are proved for solutions of the Helmholtz equation and its modified version. Also, their converses are considered along with some other properties which generalise those that guarantee harmonicity.
This article is concerned with the parabolic Monge-Amp\`ere equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of…
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We…
The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).
It is discussed that Zeeman's theorem can be directly obtained from Liouville's theorem if we assume sufficient differentiability.
We study global convex solutions of the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a…
In this paper we prove the Liouville type theorem for stable at infinity solutions of the following equation $$\Delta_{m}^{3}u =|u|^{\theta-1}u\;\;\; \mbox{in}\,\, \mathbb{R}^N,$$ for $1<m-1<\theta<\theta_{s, m}:=\frac{N(m-1)+3m }{N-3m}.$…
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
We consider the fractional generalizations of Liouville equation. The normalization condition, phase volume, and average values are generalized for fractional case.The interpretation of fractional analog of phase space as a space with…
We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples…
In this paper, we establish a complete Liouville--type hierarchy for polyharmonic functions on Riemannian manifolds with nonnegative Ricci curvature. Extending Yau's classical result for harmonic functions and our recent biharmonic…
We investigate the connection between the linear harmonic oscillator equation and some classes of second order nonlinear ordinary differential equations of Li\'enard and generalized Li\'enard type, which physically describe important…
Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the…