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My Apologies

My Apologies. Joint work with Gui-Qiang Chen, Monica Torres. The Problem. The answer is Yes – Main Result. When F is assumed to be only bounded, measurable We can’t expect the same answer. What measures are admissible?. One of the Fundamental results of GMT . Sets of Finite Perimeter.

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My Apologies

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  1. My Apologies

  2. Joint work with Gui-Qiang Chen, Monica Torres

  3. The Problem

  4. The answer is Yes – Main Result

  5. When F is assumed to be only bounded, measurableWe can’t expect the same answer

  6. What measures are admissible?

  7. One of the Fundamental results of GMT

  8. Sets of Finite Perimeter

  9. The Density Topology

  10. The Density Topology – The Union of uncountably many measurable sets Is still measurable

  11. The measure-theoretic normal to an arbitrary measurable set

  12. The Set E lies on one side of the plane determined by the normal

  13. The Reduced Boundary of E \

  14. A Pathological Example

  15. More of Example – Swiss Cheese

  16. Example of F in 3-space

  17. Fuglede: The notion of “almost every set”

  18. Ties to potential theory and Complex analysis

  19. A Slight Diversion – The Coarea Formula

  20. Fuglede’s result in a more meaningful setting

  21. The Normal Trace of F on an arbitrary set

  22. Preliminary result

  23. More of the Hypothesis

  24. Proof of the Result

  25. Consider fields F that are bounded and for which the divergence is a signed measure

  26. The Divergence Theorem for “almost every set” Fuglede

  27. Proof of Fuglede’s Result – Quick Sketch

  28. The Main Problem

  29. Definition of Normal Trace

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