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My Apologies

This work explores the complex concept of area through competing definitions and solutions in Sobolev spaces, addressing pathological cases and offering hope for solving challenging problems. The proof, based on topological and analytical insights, presents a comprehensive approach to understanding area in a mathematical context.

sean-sellers
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My Apologies

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  1. My Apologies

  2. Thank You

  3. Some Philosophy

  4. Some Philosophy -- Continued

  5. Some Philosophy -- Continued

  6. Some Philosophy -- Continued

  7. Some Philosophy -- Continued

  8. The Problem

  9. Some Basic Observations

  10. Some Basic Observations -- Continued

  11. Our problem demands that M be pathological

  12. A surface can have finite Lebesgue area and still occupy positive measure

  13. Lebesgue Area of a Surface

  14. What Is Area?

  15. Competing definitions of Area

  16. Definitive Work on Area: “Currents and Area”

  17. Sobolev functions can have “small” supports in the sense topology, but “large” in the sense of analysis

  18. More Pathology

  19. We have a solution for n=3, p>2.(First Proof)

  20. We have a solution for n=3, p>2. (Cont.)

  21. We have a solution for n=3, p>2. (Cont.)

  22. The Result Can Be Improved

  23. The Proof

  24. The Proof Requires the following ideas -- Cont

  25. Bagby-Gauthier Result

  26. Proof of Bagby-Gauthier result

  27. Continuity of Sobolev functions on subspaces

  28. The proof of Bagby-Gauthier is concluded

  29. The Main Result

  30. The proof offers some hope for solving the problem inits greatest generality

  31. Brief Outline of the Proof -- Notation

  32. Recall Two Equivalent Definitions of Sobolev Space

  33. The Precise Representative of a Sobolev Function

  34. Quasi-Continuity & The Coarea Formula

  35. The Fibers of Q are Horizontal (m+1)-planes

  36. A Sobolev Function is a continuous Sobolev function on a.e. Fiber of Q

  37. The Basic Idea

  38. The Basic Idea – Cont

  39. Linked Spheres in Full Generality

  40. Topological Degree

  41. (n-1)-manifolds cannot contain linked spheres

  42. The Main Theorem -- Concluded

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