A Mini-course on EllipticâHyperbolic Equations - ICMS

More documents

Recommendations

Info

Local canonical forms for linear equationsThe two canonical forms for linear, second-order elliptic–hyperbolicequations were discovered by Francesco Tricomi (left) and MariaCinquini-Cibrario (right).
For most of the past century, the study of linear elliptic–hyperbolicequations was generally identified with the study of the Tricomiequationyu xx + u yy = 0.Typical domain: Ω ⊂⊂ R 2 bounded by a smooth Jordan curve γin the upper half plane and the characteristic linesΓ 1 : x − 2 3 (−y)3/2 = −1, x < 0,in the lower half-plane.Γ 2 : x + 2 3 (−y)3/2 = 1, x > 0,
Page 1 and 2: A Mini-cou
Page 3 and 4: Part I. Orientation◮ Why study th
Page 5 and 6: But, for example:◮ Separating var
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: Given a function K (x, y) defined i
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

A Mini-course on EllipticâHyperbolic Equations - ICMS

Create successful ePaper yourself

Delete template?

Save as template?

A Mini-course on EllipticâHyperbolic Equations - ICMS