A Mini-course on EllipticâHyperbolic Equations - ICMS

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Geometric interpretation of the (linear) Lavrent’ev–BitsadzeequationLu = sgn[y]u xx + u yy = 0,g is Euclidean above the x-axis:with signature (+, +)ds 2 = dx 2 + dy 2 ,and Minkowskian below the x-axis:with signature (+, −) .In general:ds 2 = dx 2 − dy 2 ,◮ elliptic operators ⇐⇒ Riemannian (“Euclidean”) metrics◮ hyperbolic operators ⇐⇒ semi-Riemannian (typically,Lorentzian) metrics.
Four features of the geometric approach:◮ The type of a linear second-order equation is not a function ofthe associated linear operator at all, as that operator is alwaysthe Laplace–Beltrami operator. Instead, the type of theequation is a feature of the metric tensor on the underlyingpseudo-Riemannian manifold.◮ Any change in signature which results in a change in sign ofthe metric determinant g will change the Laplace–Beltramioperator on the metric from elliptic to hyperbolic type.◮ In order to decide which boundary value problems are naturalfor a second-order linear elliptic–hyperbolic equation, oneshould study the geometry of the underlying metric.◮ Any curve on which the change of type occurs will necessarilyrepresent a singularity of the metric tensor, as g will vanishalong that curve. However, none of the terms in the equationitself need blow up on this curve. (An example is the waveequation on extended P 2 .)
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: Geometric approach: replace the cla
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 EllipticâHyperbolic Equations - ICMS

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A Mini-course on EllipticâHyperbolic Equations - ICMS