Related papers: Mean value property for nonharmonic functions

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are H\"older continuous for any…

We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving…

Several mean value identities for harmonic and panharmonic functions are reviewed along with the corresponding inverse properties. The latter characterize balls, annuli and strips analytically via these functions.

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in…

Let $S$ be a Damek-Ricci space equipped with the Laplace-Beltrami operator $\Delta$. In this paper we characterize all eigenfunctions of $\Delta $ through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to…

Using, as main tool, the convergence theorem for discrete martingales and the mean value property of harmonic functions we solve, a particular case of, Dirichlet problem.

This article provides a novel and simple range description for the spherical mean transform of functions supported in the unit ball of an odd dimensional Euclidean space. The new description comprises a set of symmetry relations between the…

In this paper the authors investigate a power mean inequality for a special function which is defined by the complete elliptic integrals.

Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. We prove that bounded functions on (M,g) satisfying the mean-value property are constant. We extend thus a result of A.…

The primary goal of this paper is to introduce bilinear analogues of uncentered spherical averages, Nikodym averages associated with spheres and the associated bilinear maximal functions. We obtain $L^p$-estimates for uncentered bilinear…

A new numerical method is developed to approximate the solution of Laplace's equation in the exterior of the sphere with a strongly nonlinear boundary value of oblique type. A functional analysis attempt to solve this type of boundary…

A sufficient condition for a cluster point of a planar harmonic function to be an asymptotic value is given, based on a partitioning into regions of constant valence. A sufficient condition for the cluster set of a planar harmonic function…

In this paper we deal with the problem of recovering functions from their spherical mean transform $\mathcal{R}$, which integrates functions on circles in the plane, in case where the centers of the circles of integration are located on a…

A simplex in n dimensions is defined by the usual (n+1) linear inequality constraints in n dimensions. Here we consider simplexes which are bounded sets. The harmonic center has been defined earlier for polytopes in general. A relationship…

The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lam\'e-Wangerin functions called internal and external bi-cyclide harmonics. An…

In this paper we prove a criterion for plurisubharmonic functions in terms of integral mean by complex ellipsoids. Moreover, by using the criterion we prove an analogue of Blaschke-Privalov theorem for plurisubharmonic functions.

The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to…

We construct families of smooth functions $H\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that the Euclidean $(n+1)$-space is completely filled by not necessarily round hyperspheres of mean curvature $H$ at every point.

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincar\'e inequality. The notion of harmonicity is based on the Dirichlet…