Transcription of Partial Differential Equations - uni-leipzig.de
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Partial Differential Equations - uni-leipzig.de
Partial Differential EquationsLecture NotesErich MiersemannDepartment of MathematicsLeipzig UniversityVersion October, 20122 Contents1 Examples .. Equations from variational problems .. Ordinary differential Equations .. Partial differential Equations .. Exercises .. 222 Equations of first Linear Equations .. Quasilinear Equations .. A linearization method .. Initial value problem of Cauchy .. Nonlinear Equations in two variables .. Initial value problem of Cauchy .. Nonlinear Equations inRn.. Hamilton-Jacobi theory .. Exercises .. 593 Linear Equations of second order .. Normal form in two variables .. Quasilinear Equations of second order .. Quasilinear elliptic Equations .. Systems of first order.
12 CHAPTER 1. INTRODUCTION Assume u ∈ C1, then vη = 1 2 (ux −uy). If ux = uy, then vη = 0 and vice versa, thus v = w(ξ) are solutions for arbitrary C1-functions w(ξ). Consequently, we have a large class of solutions of the original partial differential equation: u = w(x + y) with an arbitrary C1-function w. 3.
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