A Mini-course on EllipticâHyperbolic Equations - ICMS

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3. The presence of an elliptic region can exert a regularizing effecton the hyperbolic region (e.g., elliptic–hyperbolic models ofcaustics in which boundary data can be prescribed in the ellipticregion, which surrounds the hyperbolic region), making theanalogy to nonlinear hyperbolic pde less apt.4. Few experts in nonlinear waves would prefer not to have a goodlinear theory of hyperbolic equations.5. An understanding of the linear theory is necessary in order tosolve nonlinear partial differential equations locally byimplicit-function methods:
Denote by Ω a domain of R n and consider the solvability of anonlinear partial differential equation F (u) = f , where f lies in thefunction space H k (Ω). If D p is a fixed open set of H p , then wewant to solve this equation for u ∈ D p . It is generally possible tofind an approximate solution u 0 such that F (u 0 ) is “close” to f insome useful sense. The goal is to use some version of the ImplicitFunction Theorem to perturb u 0 locally into a local solution to theequation F (u) = f . The solvability of the nonlinear equationdepends critically on the invertibility at u 0 of the linear operatorF ′ (u) v = d dt |t=0F (u + tv) .(There are technical problems connected with this method.) Twoimportant classes of nonlinear elliptic–hyperbolic problems ingeometry have been attacked in this way.
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: 1. Elliptic-hyperbolic continuity e
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