A Mini-course on EllipticâHyperbolic Equations - ICMS

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Three limitations of the hodograph linearization◮ It requires that the equation be of a very special form and isnot robust to small changes in that form.◮ The hodograph transformation may be singular. (Butgenerally, any singular set on the elliptic region of theequations must consist of isolated points.)◮ Whereas the hodograph mapping replaces a quasilinearequation by a linear one, it tends to replace linear boundaryconditions by nonlinear ones. In general this restricts theboundary conditions that can be applied in this method torelatively simple examples.Example: The vanishing of a solution to the Magnanini–Talentiequation (8) on a circle in the hodograph plane pulls back, under ahodograph transformation, to a constant value of the solution onthe corresponding circle in the physical plane.
The partial hodograph transformationThis is a(nother) very old method that has attracted new interestrecently. Instead of switching all the dependent variables in theequation with all the independent variables as in the classicalhodograph method, In this variant we switch just one. (In theory,switching any number of dependent variables that is fewer than allof them will result in a partial hodograph transformation.)This is not a linearization method. The original idea, due toFriedrichs [Math. Ann. 109 (1934)], was to use this method totransform a hypersurface on which a solution vanishes into a flatsurface in new coordinates. More generally, this idea can be usedto change a free boundary problem into one with a fixed boundary.That use of the method was apparently introduced, in the contextof elliptic theory, by Kinderlehrer and Nirenberg [Annali Sc. Norm.Sup. Pisa 4 (1977)].
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 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: Making the substitutions p = θ x ,
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