Mean Value Formula Research Articles

An asymptotic expansion for the fractional p-Laplacian and for gradient-dependent nonlocal operators

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well-known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to the [Formula: see text]-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the nonlocal, nonlinear mean value kernel, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) [Formula: see text]-Laplacian (for [Formula: see text]) and to other gradient-dependent nonlocal operators.

ON MAX–MIN MEAN VALUE FORMULAS ON THE SIERPINSKI GASKET

In this paper, we study solutions to the max–min mean value problem [Formula: see text] in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula [Formula: see text] is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and [Formula: see text] is the set of vertices that are adjacent to [Formula: see text] at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions.

The Harmonicity of Slice Regular Functions

In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over {mathbb {H}}. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over {mathbb {H}} (analogous to an holomorphic function over {mathbb {C}}) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over {mathbb {H}}? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over {mathbb {H}} and a Jensen’s Formula for semi-regular ones.

A sharp stability result for the Gauss mean value formula

We prove a quantitative stability result for the Gauss mean value formula. We also show by an example that the estimate proved here is sharp.

English

English

Stability of the mean value formula for harmonic functions in Lebesgue spaces

Let D be an open subset of $${\mathbb {R}}^n$$ with finite measure, and let $$x_0 \in D$$ . We introduce the p-Gauss gap of D w.r.t. $$x_0$$ to measure how far are the averages over D of the harmonic functions $$ u \in L^p(D)$$ from $$u(x_0)$$ . We estimate from below this gap in terms of the ball gap of D w.r.t. $$x_0$$ , i.e., the normalized Lebesgue measure of $$D \setminus B$$ , being B the biggest ball centered at $$x_0$$ contained in D. From these stability estimates of the mean value formula for harmonic functions in $$L^p$$ -spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space $$W^{1,p'}$$ , where $$p'$$ is the conjugate exponent of p.

Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair h_1, h_2 of zonal harmonic functions such that f=h_1-bar{x} h_2. We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

Maximum-Entropy Based Estimates of Stress and Strain in Thermoelastic Random Heterogeneous Materials

Mean-field methods are a common procedure for characterizing random heterogeneous materials. However, they typically provide only mean stresses and strains, which do not always allow predictions of failure in the phases since exact localization of these stresses and strains requires exact microscopic knowledge of the microstructures involved, which is generally not available. In this work, the maximum entropy method pioneered by Kreher and Pompe (Internal Stresses in Heterogeneous Solids, Physical Research, vol. 9, 1989) is used for estimating one-point probability distributions of local stresses and strains for various classes of materials without requiring microstructural information beyond the volume fractions. This approach yields analytical formulae for mean values and variances of stresses or strains of general heterogeneous linear thermoelastic materials as well as various special cases of this material class. Of these, the formulae for discrete-phase materials and the formulae for polycrystals in terms of their orientation distribution functions are novel. To illustrate the theory, a parametric study based on Al-Al2O3 composites is performed. Polycrystalline copper is considered as an additional example. Through comparison with full-field simulations, the method is found to be particularly suited for polycrystals and materials with elastic contrasts of up to 5. We see that, for increasing contrast, the dependence of our estimates on the particular microstructures is increasing, as well.

Dirac Operators with Gradient Potentials and Related Monogenic Functions

We investigate some properties of solutions to Dirac operators with gradient potentials. Solutions to Dirac operators with gradient potentials are called monogenic functions with respect to the potential functions. We establish a Borel–Pompeiu formula, and obtain Cauchy integral formula and mean value formula about such functions. Based on integral formulas, we prove some geometric properties of monogenic functions with respect to the potential functions and construct some integral transforms. The boundedness of integral transforms in Holder space is given. We prove the Plemelj–Sokhotski formula and the Painleve theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the potential functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert problem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem.

Some new hybrid power mean formulae of trigonometric sums

We apply the analytic method and the properties of the classical Gauss sums to study the computational problem of a certain hybrid power mean of the trigonometric sums and to prove several new mean value formulae for them. At the same time, we also obtain a new recurrence formula involving the Gauss sums and two-term exponential sums.

Two-Point Mean Value Formulas

We consider the mean value theorem for the wave equation in the Euclidean space to be a starting point to produce other mean value theorems for the wave equations on a sphere and in a space. We use transformations connecting the wave equation in the Euclidean space and the wave equation in non-Euclidean spaces. Also, we prove two-point mean value formulas for elliptic equations in the Lobachevskii space and on a sphere.

Dirichlet-to-Neumann Maps on Trees

In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a "gradient" with a "normal vector" and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann map is given by $g\mapsto cg'$ (here $c$ is an explicit constant). Notice that this is a local operator of order one. We also consider linear undirected mean value formulas (taking into account not only the successors but the ancestor and the successors of a given node) and prove a similar result. For this kind of mean value formula we include some existence and uniqueness results for the associated Dirichlet problem. Finally, we give an alternative definition of the Dirichlet-to-Neumann map (taking into account differences along a given branch of the tree). With this alternative definition, for a certain range of parameters, we obtain that the Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the classical Laplacian in the Euclidean space).

Mean value formulas for Ornstein–Uhlenbeck and Hermite temperatures

We obtain explicit mean value formulas for the solutions of the diffusion equations associated with the Ornstein–Uhlenbeck and Hermite operators. From these, we derive various useful properties, such as maximum principles, uniqueness theorems and Harnack-type inequalities.

Lax and Phillips transformations for a sphere and two-point mean value formulas

We consider transformations, connecting the wave equation in Euclidean space and the wave equation over a sphere that has a metric of a hemisphere immersed in Euclidean space. Using these transformations, we prove a two-point mean value theorem for an elliptic equation on a sphere.

Some Remarks on Harmonic Functions in Minkowski Spaces

We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that in a Minkowski space, if all harmonic functions have the mean value property or any function satisfying the mean value formula must be a harmonic function, then the Minkowski space is Euclidean.

On the generalized circle problem for a random lattice in large dimension

In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random functiont↦12f(n)Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.

Mean values of derivatives of L-functions in function fields: III

AbstractIn this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

NONLINEAR MEAN-VALUE FORMULAS ON FRACTAL SETS

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem [Formula: see text] in the Sierpiński gasket with prescribed values [Formula: see text], [Formula: see text] and [Formula: see text] at the three vertices of the first triangle. For this problem we show existence and uniqueness of a continuous solution and analyze some properties like the validity of a comparison principle, Lipschitz continuity of solutions (regularity) and continuous dependence of the solution with respect to the prescribed values at the three vertices of the first triangle.

Sums of weighted averages of gcd-sum functions

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]

A Mean Value Formula and a Liouville Theorem for the Complex Monge–Ampère Equation

Abstract In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge–Ampère equation on product manifolds.