Math 2.0

Partial differential equations Spectral theory and eigenvalue problems Problem

Pompeiu problem

Builds on Lebesgue measure

Problem

Let n≥2n \ge 2 and let K⊆ℝnK \subseteq \R^n be a bounded set with nonempty interior. The set KK has the Pompeiu property if the only continuous function f ⁣:ℝn→ℂf \colon \R^n \to \C with ∫σ(K)f dx=0for every rigid motion σ of ℝn\int_{\sigma(K)} f \, dx = 0 \quad \text{for every rigid motion } \sigma \text{ of } \R^n is f=0f = 0. Here a rigid motion is a composition of a rotation and a translation, and the integral is against Lebesgue measure. Determine which sets have the Pompeiu property; in particular, decide whether every bounded simply connected Lipschitz domain without the Pompeiu property is a ball.

Remarks

A ball does not have the Pompeiu property. For the unit ball in ℝn\R^n and a suitable nonzero vector aa, the function f(x)=sin⁡(a⋅x)f(x) = \sin(a \cdot x) has integral 00 over every translate of the ball, and rotations map balls to balls. Squares and other polygons are known to have the Pompeiu property. By a theorem of Williams (1976), for smooth contractible domains the conjectured answer is equivalent to Schiffer's conjecture on an overdetermined Neumann eigenvalue problem.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectPompeiu problem, problem, no evidence levelPompeiu problemproblem
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  1. Borel sigma-algebra concept
  2. Euclidean space object
  3. Outer measure concept
  4. Lebesgue measure object
  5. Pompeiu problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Pompeiu_problem
About this page
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MSC35N25, 44A12, 35R30
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AboutSchiffer's conjecture
UsesLebesgue measure
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