Mean Value Property Research Articles

Weighted means and an analytic characterization of discs

Weighted means are obtained for solutions of the two-dimensional Helmholtz and modified Helmholtz equations and also for harmonic functions. The presence of a logarithmic weight reduces the coefficient in the last two mean value identities. A new theorem characterizing disks in the Euclidean plane R 2 \mathbb {R}^2 analytically is proved; it is based on the weighted mean value property of solutions to the modified Helmholtz equation.

Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets

AbstractWe give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $$\alpha $$ α -stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary.

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

A unified framework for perceived magnitude and discriminability of sensory stimuli

The perception of sensory attributes is often quantified through measurements of sensitivity (the ability to detect small stimulus changes), as well as through direct judgments of appearance or intensity. Despite their ubiquity, the relationship between these two measurements remains controversial and unresolved. Here, we propose a framework in which they arise from different aspects of a common representation. Specifically, we assume that judgments of stimulus intensity (e.g., as measured through rating scales) reflect the mean value of an internal representation, and sensitivity reflects a combination of mean value and noise properties, as quantified by the statistical measure of Fisher information. Unique identification of these internal representation properties can be achieved by combining measurements of sensitivity and judgments of intensity. As a central example, we show that Weber’s law of perceptual sensitivity can coexist with Stevens’ power-law scaling of intensity ratings (for all exponents), when the noise amplitude increases in proportion to the representational mean. We then extend this result beyond the Weber’s law range by incorporating a more general and physiology-inspired form of noise and show that the combination of noise properties and sensitivity measurements accurately predicts intensity ratings across a variety of sensory modalities and attributes. Our framework unifies two primary perceptual measurements—thresholds for sensitivity and rating scales for intensity—and provides a neural interpretation for the underlying representation.

Nonlinear asymptotic mean value characterizations of holomorphic functions

Starting from a characterization of holomorphic functions in terms of a suitable mean value property, we build some nonlinear asymptotic characterizations for complex-valued solutions of certain nonlinear systems, which have to do with the classical Cauchy-Riemann equations. From these asymptotic characterizations, we derive suitable asymptotic mean value properties, which are used to construct appropriate vectorial dynamical programming principles. The aim is to construct approximation schemes for the so-called contact solutions, recently introduced by N. Katzourakis, of the nonlinear systems here considered.

Remarks about the mean value property and some weighted Poincaré-type inequalities

Remarks about the mean value property and some weighted Poincaré-type inequalities

On mean value properties involving a logarithm-type weight

On mean value properties involving a logarithm-type weight

Alt–Caffarelli–Friedman monotonicity formula and mean value properties in Carnot groups with applications

Alt–Caffarelli–Friedman monotonicity formula and mean value properties in Carnot groups with applications

Asymptotic mean value properties for the elliptic and parabolic double phase equations

Asymptotic mean value properties for the elliptic and parabolic double phase equations

One-radius theorem for harmonic tempered distributions

Abstract We show that the “one-radius” spherical mean value property is sufficient to characterize harmonic tempered distributions subject to the classical Laplace operator.

Boundary-free estimators of the mean residual life function for data on general interval

We propose two new kernel-type estimators of the mean residual life function m X of bounded or half-bounded interval-supported distributions. Though not as severe as the boundary problems in kernel density estimation, eliminating the boundary bias problems that occur in the naive kernel estimator of the mean residual life function is needed. In this article, we utilize the property of bijective transformation. Furthermore, our proposed methods preserve the mean value property, which cannot be done by the naive kernel estimator. Some simulation results showing the estimators’ performances and real data analysis will be presented in the last part of this article.

Panharmonic Functions: Mean Value Properties and Related Topics

Recently obtained results about m-dimensional panharmonic functions, i.e., solutions to the modified Helmholtz equation, are presented in a systematic form. They concern various mean value properties. Their corollaries are considered along with converse and inverse of these properties, and relations between panharmonic and harmonic functions.

Inverse mean value property of solutions to the modified Helmholtz equation

A theorem characterizing balls in the Euclidean space R m \mathbb {R}^m analytically is proved. For this purpose, positive solutions of the modified Helmholtz equation are applied instead of harmonic functions used in previous results. The resulting Kuran type theorem involves the volume mean value property of solutions to this equation. Other plausible inverse mean value properties of these solutions are discussed.

Tug-of-war with Kolmogorov

We introduce a new class of strongly degenerate nonlinear parabolic PDEs((p−2)Δ∞,XN+ΔX)u(X,Y,t)+(m+p)(X⋅∇Yu(X,Y,t)−∂tu(X,Y,t))=0,(X,Y,t)∈Rm×Rm×R, p∈(1,∞), combining the classical PDE of Kolmogorov and the normalized p-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games introduced approximate solutions to the stated PDE when the parameter that controls the size of the possible steps goes to zero. Existence and uniqueness of viscosity solutions to the Dirichlet problem is established. The asymptotic mean value property, the associated games and the geometry underlying the Dirichlet problem, all reflect the family of dilation and the Lie group underlying operators of Kolmogorov type, and this makes our setting different from the context of standard parabolic dilations and Euclidean translations applicable in the context of the heat operator and the normalized parabolic infinity Laplace operator.

FRACTIONAL WEIGHTED SPHERICAL MEAN AND MAXIMAL INEQUALITY FOR THE WEIGHTED SPHERICAL MEAN AND ITS APPLICATION TO SINGULAR PDE

In this paper we establish a mean value property for the functions which is satisfied to Laplace–Bessel equation. Our results involve the generalized divergence theorem and the second Green’s identities relating the bulk with the boundary of a region on which differential Bessel operators act. Also we design a fractional weighted mean operator, study its boundedness, obtain maximal inequality for the weighted spherical mean and get its boundedness. The connection between the boundedness of the spherical maximal operator and the properties of solutions of the Euler–Poisson–Darboux equation with Bessel operators is given as an application.

Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.

Inverse Mean Value Property of Metaharmonic Functions

A new analytical characterization of balls in the Euclidean space ℝm is obtained. Previous results of this kind involved either harmonic functions or solutions to the modified Helmholtz equation (both have positive fundamental solutions), whereas solutions to the Helmholtz equation are used here. This is achieved at the expense of a restriction imposed on the size of admissible domains—a feature absent in the inverse mean value properties known previously.

Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

AbstractWe consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

Mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane)

mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane)

Asymptotic mean value property for eigenfunctions of the Laplace–Beltrami operator on Damek–Ricci spaces

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 infinity.