Elliptic operators and boundary problems

This course develops elliptic operators and boundary problems from their analytic foundations. It begins with metric spaces, Banach and Hilbert spaces, Fourier transformation and spectral arguments, then builds symbol estimates, composition and inversion, regularity, Fredholm operators, trace formulas and index calculations. The later lessons treat operator systems, boundary modes, boundary measurements and reductions, including the Bott operator, Weyl calculus, elliptic complexes and adapted hypoelliptic Fredholm arguments.

The collection contains all 56 complete lesson texts. Their HTML readings preserve every original formula as a TeX annotation beside MathML, with complete arguments, examples and solved exercises. Current PDF and TeX readers are available for 56 lessons; every lesson can be read in HTML and in its matching PDF.

The Dirichlet lesson includes the complete boundary regularity arguments, sharp spectral cutoff kernels, local trace formulas and singular multiplication domains. Its matching PDF and TeX reader is available below.

Every mathematical result used is proved in its lesson or an earlier programme lesson. Free sources only: freely accessible author versions, papers, lecture notes and open textbooks provide human comparisons and citations. A citation never replaces a proof. Paywalled works are not mathematical sources, proof templates or public references.

Reading order

1. Metric and topological foundations · Markdown · PDF · TeX
   Complete HTML
2. Banach estimates, quotient spaces and compact parameter arguments · Markdown · PDF · TeX
   Complete HTML
3. Fourier transforms, finite spectra and convex separation · Markdown · PDF · TeX
   Complete HTML
4. Polynomial and contour interfaces for stable boundary models · Markdown · PDF · TeX
   Complete HTML
5. Spectral measures with the original operator domain retained · Markdown · PDF · TeX
   Complete HTML
6. Localizing symbols with moving metrics · Markdown · PDF · TeX
   Complete HTML
7. Quadratic Fourier multipliers at a moving scale · Markdown · PDF · TeX
   Complete HTML
8. Two measuring scales, one Weyl product · Markdown · PDF · TeX
   Complete HTML
9. From Weyl symbols to operators and changes of coordinates · Markdown
   Complete HTML; matching PDF and TeX · PDF · TeX
10. Symbols, operators and Sobolev scales · Markdown · PDF · TeX
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11. Positivity through a moving family of scalar probes · Markdown · PDF · TeX
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12. When a moving symbol scale controls an operator · Markdown · PDF · TeX
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13. When a nonnegative scalar symbol acquires a negative part · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
14. Singularities along a submanifold and smooth boundary passage · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
15. Detecting regularity without choosing coordinates · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
16. Local inverses and distance-weighted elliptic estimates · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
17. Curved weights and the directions in which support can end · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
18. Detecting a solution from infinite-order silence at one point · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
19. From local energy to global divergence equations · Markdown · PDF · TeX
    Complete HTML
20. Positive energy and vanishing at the far end of space · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
21. Inverting mixed symbols without commuting matrix factors · Markdown · PDF · TeX
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22. Polynomial inverse expansion with ordered matrix coefficients · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
23. Composition of mixed symbols with two different remainder estimates · Markdown · PDF · TeX
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24. Mixed symbols on every real two-parameter Sobolev scale · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
25. Totally characteristic operators on the half space · Markdown · PDF · TeX
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26. Global boundary operators, compressed wave fronts, and normal extension · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
27. Changing an interior frame to extend an invertible matrix · Markdown · PDF · TeX
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28. Finite defects under perturbation · Markdown · PDF · TeX
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29. Traces that survive passage to cohomology · Markdown · PDF · TeX
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30. Weyl kernels, operator traces, and a finite trace-class test · Markdown · PDF · TeX
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31. Scaled Weyl parametrices and the surviving differential degree · Markdown · PDF · TeX
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32. Radial compression and index transport for product-metric symbols · Markdown · PDF · TeX
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33. Relative Weyl projectors and the Chern cutoff form · Markdown · PDF · TeX
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34. From ordered Weyl errors to the exterior index form · Markdown · PDF · TeX
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35. The matrix Weyl index in the original phase coordinates · Markdown · PDF · TeX
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36. Alternation and the index under linear phase changes · Markdown · PDF · TeX
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37. Symbols, finite defects, and the index on a closed manifold · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
38. When finite defects force one-sided ellipticity · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
39. Nonelliptic Fredholm operators on their exact adapted spaces · Markdown · PDF · TeX
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40. Elliptic complexes, diagonal traces, and fixed points · Markdown · PDF · TeX
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41. The Bott operator, suspension, and reduction of the index to Euclidean space · Markdown · PDF · TeX
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42. Characteristic classes, Bott normalization, fixed points, and external index theories · Markdown · PDF · TeX
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43. Building a local inverse from radial singularities · Markdown · PDF · TeX
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44. Causal kernels, initial data, and short-time geometry · Markdown · PDF · TeX
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45. A wave kernel that cancels at a curved boundary · Markdown · PDF · TeX
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46. Stable modes and the algebra of boundary data · Markdown · PDF · TeX
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47. Boundary energy, local inverses, and harmonic data · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
48. Cauchy data from jumps and residues · Markdown · PDF · TeX
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49. Solving an elliptic system from compatible boundary measurements · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
50. Fredholm boundary problems with first-order Calderón defects · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
51. Doubling a boundary problem and computing its index · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
52. Boundary wave fronts for elliptic systems · Markdown
    Complete HTML; matching PDF and TeX · PDF · TeX
53. Reducing first-order boundary data to a split trace · Markdown · PDF · TeX
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54. Reducing arbitrary-order boundary problems to first order · Markdown · PDF · TeX
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55. Reducing arbitrary boundary data to a boundary system · Markdown · PDF · TeX
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56. Dirichlet realizations, spectral projectors, and local extensions — Markdown · Complete HTML proofs. · PDF · TeX