A Mini-course on EllipticâHyperbolic Equations - ICMS

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Naive discussion of equation type:Physically, what do solutions of partial differential equations do?In general, they◮ propagate as waves◮ diffuse as heat◮ do nothing at all (e.g., “oscillate”)This leads to classification of the corresponding equations as,respectively◮ hyperbolic◮ parabolic◮ elliptic
But, for example:◮ Separating variables in Laplace’s equation in toroidalcoordinates produces a partial differential equation which iselliptic inside the unit circle and hyperbolic on theR 2 -complement of the unit circle;[B. Riemann, Partielle Differentialgleichungen, 1861; also C.Neumann, 1864; W. M Hicks, 1881; E. Heine, 1881; A. B.Basset, 1893]◮ the equations for a steady, irrotational, isentropic,compressible flow are elliptic for velocities below the speed ofsound in the medium, and hyperbolic at velocities above thespeed of sound;[H. Bateman, Proc. R. Soc. London Ser. A 125 (1929)]
Page 1 and 2: A Mini-cou
Page 3: Part I. Orientation◮ Why study th
Page 7 and 8: ◮ the governing equations for ide
Page 9 and 10: ◮ the Laplace-Beltrami equation o
Page 11 and 12: ◮ In certain models for plasma fl
Page 13 and 14: ◮ The relevant equation for the e
Page 15 and 16: Adopting the “differential polyno
Page 17 and 18: Example of a first-order elliptic-h
Page 19 and 20: We obtain from the first-order syst
Page 21 and 22: Geometric approach: replace the cla
Page 23 and 24: Four features of the geometric appr
Page 25 and 26: 1. Elliptic-hyperbolic continuity e
Page 27 and 28: Denote by Ω a domain of R n and co
Page 29 and 30: Given a function K (x, y) defined i
Page 31 and 32: For most of the past century, the s
Page 33 and 34: Tricomi’s assistant Cinquini-Cibr
Page 35 and 36: M.V. Keldysh had nothing to do with
Page 37 and 38: Equations of Keldysh type can be ch
Page 39 and 40: Why do equations of Keldysh type ha
Page 41 and 42: Let ξ = K (x, y) and η = λ (x, y
Page 43 and 44: Moreover the Implicit Function Theo
Page 45 and 46: If a system of equations has the fo
Page 47 and 48: The coordinate systems (p, q) and (
Page 49 and 50: Making the substitutions p = θ x ,
Page 51 and 52: The partial hodograph transformatio
Page 53 and 54: Example for an elliptic-hyperbolic
Page 55 and 56:
Because the nonlinear Lavrent’ev-
Page 57 and 58:
In order to fix the boundary, we no
Page 59 and 60:
Example 2: Find an L 2 solution to
Page 61 and 62:
Open boundary conditions: Data are
Page 63 and 64:
Geometries of canonical domains:Geo
Page 65 and 66:
Ingredients:1. A one-parameter fami
Page 67 and 68:
Moreover, whenever a domain is star
Page 69 and 70:
Example: the cold plasma model(x
Page 71 and 72:
TheoremLet Ω be star-shaped with r
Page 73 and 74:
Idea of the proof: Integral variant
Page 75 and 76:
If we find such a, b, and c, then w
Page 77 and 78:
Suppose instead that we prove the f
Page 79:
Instead of the L 2 solution which r
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A Mini-course on EllipticâHyperbolic Equations - ICMS