Pompeiu Property Research Articles

Univalent Functions and the Pompeiu Problem

In this paper we prove a result on the Pompeiu problem. If the Schwarz function $\Phi$ of the boundary of a simply-connected domain $\Omega \subset {\mathbb {R}^2}$ extends meromorphically into a certain portion $E$ of $\Omega$ with a pole at some point ${z_0} \in E$, then $\Omega$ has the Pompeiu property unless $\Phi$ is a Möbius transformation, in which case $\Omega$ is a disk.

The Cauchy problem in 𝐶^{𝑁} for linear second order partial differential equations with data on a quadric surface

By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray’s local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points. Finally the latent singularities are sets which intersect the initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of R N {\mathbb {R}^N} . As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.

Univalent functions and the Pompeiu problem

In this paper we prove a result on the Pompeiu problem. If the Schwarz function Φ \Phi of the boundary of a simply-connected domain Ω ⊂ R 2 \Omega \subset {\mathbb {R}^2} extends meromorphically into a certain portion E E of Ω \Omega with a pole at some point z 0 ∈ E {z_0} \in E , then Ω \Omega has the Pompeiu property unless Φ \Phi is a Möbius transformation, in which case Ω \Omega is a disk.

New Results on the Pompeiu Problem

Let ${p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}}$, $w \in \mathbb {C}$, $N \in \mathbb {N}$, be a polynomial with complex coefficients. In this paper we prove that if $D \subset {\mathbb {R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $s \in [ - \pi ,\pi ]$, then $D$ has the Pompeiu property unless $N = 1$ and ${p_1}(w) = {a_1}w + {a_2}$ in which case $D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.

New results on the Pompeiu problem

Let p N ( w ) = ∑ k = 0 N a k w k {p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}} , w ∈ C w \in \mathbb {C} , N ∈ N N \in \mathbb {N} , be a polynomial with complex coefficients. In this paper we prove that if D ⊂ R 2 D \subset {\mathbb {R}^2} is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by x ( s ) = x 1 ( s ) + i x 2 ( s ) = p N ( e i s ) x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}}) , s ∈ [ − π , π ] s \in [ - \pi ,\pi ] , then D D has the Pompeiu property unless N = 1 N = 1 and p 1 ( w ) = a 1 w + a 2 {p_1}(w) = {a_1}w + {a_2} in which case D D is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.

A partial solution of the Pompeiu problem

A nonempty bounded open subset D of ℝ n is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ℝ n which does not vanish identically there is a rigid motion σ of ℝ n onto itself — taking D onto σ(D) — such that the integral of f over σ(D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property. In the special case that D is the interior of a homeomorphic image of an(n−1)-dimensional sphere, the main result states that if D has a portion of an(n−1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.