My Apologies

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My Apologies

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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. - PowerPoint PPT Presentation

Transcript of My Apologies

Page 1: My Apologies

My Apologies

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Joint work with Gui-Qiang Chen, Monica Torres

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The Problem

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The answer is Yes – Main Result

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

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What measures are admissible?

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One of the Fundamental results of GMT

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Sets of Finite Perimeter

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The Density Topology

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

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The measure-theoretic normal to an arbitrary measurable set

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

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The Reduced Boundary of E

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A Pathological Example

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More of Example – Swiss Cheese

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Example of F in 3-space

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Fuglede: The notion of “almost every set”

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Ties to potential theory and Complex analysis

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A Slight Diversion – The Coarea Formula

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Fuglede’s result in a more meaningful setting

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The Normal Trace of F on an arbitrary set

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Preliminary result

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More of the Hypothesis

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Proof of the Result

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

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The Divergence Theorem for “almost every set” Fuglede

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Proof of Fuglede’s Result – Quick Sketch

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The Main Problem

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Definition of Normal Trace