Related papers: Mean value property for nonharmonic functions
On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the…
Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical…
Ultrafunctions are a particular class of functions defined on a non-Archimedean field. They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper…
An elementary general result is proved that allows for simple characterizations of well-known location/consensus functions (median, mean and center) on the n-cube. In addition, alternate new characterizations are given for the median and…
Solutions to the $n$-dimensional Laplace equation which are constant on a central quadric are found. The associated twistor description of the case $n=3$ is used to characterise Gibbons-Hawking metrics with tri-holomorphic $SL(2, \C)$…
We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the…
As shown recently [Phys. Rev. E 95, 033307 (2017)], spheroidal harmonics expansions are well suited for the external solution of Laplace's equation for a point source outside a spherical object. Their intrinsic singularity matches the line…
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible…
We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal…
The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type…
We give background which shows the connection between the mean value theorem and the obstacle problem, and then we prove that a set is a mean value set for an elliptic operator of the form $Lu := \partial_i (a^{ij}(x) \partial_j u(x))$ if…
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make…
In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and Laplace transform.
A floating hemisphere under forced harmonic oscillation at very high and very low frequencies is considered. The problem is reduced to an elliptic one, that is, the Laplace operator in the exterior domain with standard Dirichlet and Neumann…
This paper presents necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. In addition to motivating this study and identifying useful criteria for determining whether such…
This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative…
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…
We provide an asymptotic expansion for the mean-value of the logarithm of the middle prime factor of an integer, defined according to multiplicity or not, thus generalising a recent study of McNew, Pollack, and Singha Roy. This yields an…
In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the…
We write down a one-dimensional integral formula and compute large-n asymptotics for the expectation of the absolute value of the smallest component of a unit vector in n-dimensional Euclidean space. The method is general, and allows to…