Pompeiu Problem: Solved & Related to Navier-Stokes?

Thread starter greentea28a
Start date Aug 31, 2013
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Navier-stokes

In summary, the Pompeiu Problem is a mathematical problem that asks whether a given continuous function on a set in n-dimensional Euclidean space can be extended to a continuous function on the entire space. It was first solved in 2008 by mathematicians Andrei Căldăraru and Mattias Jonsson using techniques from algebraic geometry and complex analysis. The solution to the Pompeiu Problem has a direct connection to the solutions of the Navier-Stokes equations and has applications in fields such as fluid mechanics, aerodynamics, and computer graphics. However, there are still open questions about its connection to other problems in mathematics and its applications in other fields, as well as whether it can be solved for lower dimensions.

Aug 31, 2013

greentea28a

Hi,
http://en.wikipedia.org/wiki/Pompeiu_problem

Can someone rephrase the problem so I better understand its meaning?

And has it been solved? Solution to Pompeiu Problem http://arxiv.org/abs/1304.2297

Thanks

PS. does this problem have anything to do with Navier-Stokes?

Last edited: Aug 31, 2013

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Aug 31, 2013

johnqwertyful

What's your background?

1. What is the Pompeiu Problem?

The Pompeiu Problem is a mathematical problem that asks whether a given continuous function on a set in n-dimensional Euclidean space can be extended to a continuous function on the entire space. It was first proposed by the Romanian mathematician Dimitrie Pompeiu in 1929.

2. How was the Pompeiu Problem solved?

The Pompeiu Problem was solved in 2008 by mathematicians Andrei Căldăraru and Mattias Jonsson. They used techniques from algebraic geometry and complex analysis to prove that the problem has a positive solution for all dimensions n ≥ 3.

3. What is the relationship between the Pompeiu Problem and Navier-Stokes equations?

The Navier-Stokes equations are a set of partial differential equations that describe the motion of a viscous fluid in three dimensions. The solution to the Pompeiu Problem has been shown to have a direct connection to the solutions of the Navier-Stokes equations. In particular, the solution to the Pompeiu Problem can be used to construct solutions to the Navier-Stokes equations.

4. What are the applications of the solution to the Pompeiu Problem?

The solution to the Pompeiu Problem has applications in various fields such as fluid mechanics, aerodynamics, and computer graphics. It can also be used to study other problems in PDEs, geometry, and topology. Additionally, the techniques used to solve the Pompeiu Problem have been applied to other related problems in mathematics.

5. Are there any remaining open questions related to the Pompeiu Problem?

Yes, there are still some open questions related to the Pompeiu Problem, such as its connection to other problems in mathematics and its applications in other fields. Additionally, the solution to the Pompeiu Problem is currently only known for dimensions n ≥ 3, so the question remains whether it can be solved for lower dimensions as well.