Restricted Mean Value Property Research Articles

Mean Value Properties of Solutions to the Helmholtz and Modified Helmholtz Equations

Mean value properties of solutions to the m-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given. It involves the Euler–Poisson–Darboux equation. Despite the similar form of these properties for both equations, their consequences distinguish essentially. The restricted mean value property of harmonic functions is amended so that a function satisfying this property in a bounded domain of a special class solves the modified Helmholtz equation in this domain.

Restricted Mean Value Property for Balayage Spaces with Jumps

It is shown that, for α-stable processes (Riesz potentials) or—more generally—for balayage spaces with jumps, “one-radius” results for harmonicity can be obtained under fairly weak assumptions.

Liouville's theorem and the restricted mean value property in the plane

Let U be a domain in R 2 such that U c is polar and let τ be a real function on U such that 0 < r ≤ ∥ · ∥ + M. A positive numerical function f on U is called r-supermedian if, for every x ∈ U, the average of f on the disk of center x and radius r(x) is at most f(x). The purpose of this note is to give a short proof for the fact that every l.s.c. r-median function is constant.

The restricted mean value property for sublaplacians

The restricted mean value property for sublaplacians

Restricted mean value property for positive functions.

Restricted mean value property for positive functions.

A converse to the mean value theorem for harmonic functions

(A Lebesgue measure on Rd). The converse question to what extent this restricted mean value property implies harmonicity has a long history (we axe indebted to I. Netuka for valuable hints). Volterra [26] and Kellogg [20] noted first that a continuous function f on the closure U of U satisfying (*) is harmonic on U. At least if U is regular there is a very elementary proof for this fact (see Burckel [7]): Let g be the difference between f and the solution of the Dirichlet problem with boundary value f . If g#0 , say a=sup g(U)> 0, choose x6{g=a} having minimal distance to the boundary. Then (*) leads to an immediate contradiction. In fact, for continuous functions on U the question is settled for arbitrary harmonic spaces and arbitrary representing measures #x # ~ for harmonic functions. If f is bounded on U and Borel measurable the answer may be negative unless restrictions on the radius r(x) of the balls B ~ are imposed (Veech [23]): Let U = ] I , 1[, / (0 )=0 , f = i on ] -1 ,0[ , f= l on ]0, 1[, 0~B x for x?t0 (similarly in R d, d/>2)! There are various positive results, sometimes under restrictions on U, but always under restrictions on the function x~-+r(x) (Feller [9], Akcoglu and Sharpe [1], Baxter [2] and [3], Heath [17], Veech [23] and [24]). For example Heath [17] showed for arbitrary U that a bounded Lebesgue measurable function on U having the restricted mean value property (.) is harmonic provided that , for some e>0, cd(x, CU)<r(x)< (1-~)d(x, CU) holds for every x6U. Veech [23] proved that a Lebesgue measurable function f on U

On the Restricted Mean Value Property

Suppose that $u$ is continuous in the open unit disc and has the restricted mean value property. It is shown that if $u$ has finite boundary limits almost everywhere, and if $u$ possesses a harmonic majorant and minorant, the difference between which has finite radial upper limits everywhere, then $u$ is harmonic.

On the restricted mean value property

Suppose that u u is continuous in the open unit disc and has the restricted mean value property. It is shown that if u u has finite boundary limits almost everywhere, and if u u possesses a harmonic majorant and minorant, the difference between which has finite radial upper limits everywhere, then u u is harmonic.

Harmonic Functions and Mass Cancellation

If a function on an open set in ${\textbf {R}^n}$ has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius $\delta (x)$, a function of x. Under appropriate conditions on $\delta$ and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function $\delta$ is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.

Harmonic functions and mass cancellation

If a function on an open set in R n {\textbf {R}^n} has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius δ ( x ) \delta (x) , a function of x. Under appropriate conditions on δ \delta and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function δ \delta is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.

Functions Having the Restricted Mean Value Property

Functions Having the Restricted Mean Value Property

Functions possessing restricted mean value properties

A real-valued function f f defined on an open subset of R N {R^N} is said to have the restricted mean value property with respect to balls (spheres) if, for each point x x in the set, there exists a ball (sphere) with center x x and radius r ( x ) r(x) such that the average value of f f over the ball (sphere) is equal to f ( x ) f(x) . If f f is harmonic then it has the restricted mean value property. Here new conditions for the converse implication are given.

Restricted mean values and harmonic functions

A function h defined on a region R in R n {{\mathbf {R}}^n} will be said to possess a restricted mean value property if the value of the function at each point is equal to the mean value of the function over one open ball in R, with centre at that point. It is proved here that this restricted mean value property implies h is harmonic under certain conditions.