Related papers: Liouville's theorem for the generalized harmonic f…
In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady K\"ahler-Ricci soliton, then we prove that any pluriharmonic…
We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.
This paper is a study of harmonic maps from Riemannian polyhedra to (locally) non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different…
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature.…
In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\…
In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…
We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral…
Suggestions concerning the generalization of the geometric quantization to the case of nonlinear field theories are given. Results for the Liouville field theory are presented.
In this paper we classify M\"{o}bius invariant differential operators of second order in two dimensional Euclidean space, and establish a Liouville type theorem for general M\"{o}bius invariant elliptic equations.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation. This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this…
We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and…
Classical Sturm-Liouville problems of $q$-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.
We prove a generalization of the classical Poincar\'e--Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted…
Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated…
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\overrightarrow{l}+C_*\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy. \end{equation*} Here $u:…
The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezin's inequality, allows us to obtain a refined version the Li-Yau estimate for the counting…
We generalize the spherical harmonics for l=1 and give the differential equation that the generalized forms satisfy. The new forms have an obvious interpretation in the context of quantum mechanics.
The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions,…
In this survey we shall collect the main results known up to now (July 2015) regarding possible generalizations to several complex variables of the classical Leau-Fatou flower theorem in holomorphic parabolic dynamics.