A Mini-course on EllipticâHyperbolic Equations - ICMS

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A linear example [Lavrent’ev & Bitsadze, Dokl. Akad. Nauk 20(1950)]:Lu = sgn[y]u xx + u yy = 0,for which the sonic curve is the x-axis.Laplace’s equation on the upper half-planeWave equation on the lower half plane.A quasilinear example [S-X. Chen, Acta Math. Sci. 31B (2011)]:sgn[u]u xx + u yy = 0,a model for the reflection of certain shock waves in compressiblegas dynamics.In the nonlinear case the sonic curve cannot be identified withoutsolving the equation.
Example of a first-order elliptic–hyperbolic systemA scalar equation arising in, e.g., nonlinear elasticitywhere x ∈ R and t ∈ R + .u tt = σ (u x ) x, (1)Assumptions on the stress function σ that lead to a change inequation type: ∃ α, β, γ, and δ such that, e.g.,
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: Adopting the “differential polyno
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