Partial Differential EquationsSpringer Science & Business Media, 2013 M11 11 - 221 páginas These Notes grew out of a course given by the author in 1952-53. Though the field of Partial Differential Equations has changed considerably since those days, particularly under the impact of methods taken from Functional Analysis, the author feels that the introductory material offered here still is basic for an understanding of the subject. It supplies the necessary intuitive foundation which motivates and anticipates abstract formulations of the questions and relates them to the description of natual phenomena. In the present edition, only minor corrections have been made in the text. An Index and up-to-date listing of books recommended for further study have been added. Fritz John New York November 19, 1970 v TABLE OF CONTENTS Introduetion 1 CHAPl'ER I - TEE SINGLE FIRST ORDER EQUATION 1. The linear and quasi-linear equations. 6 The general first order equation for a funetion 2. of two variables. • • • • • • • • • 15 The general first order equation for a funetion 3. of n independent variables. • • • • • 37 CHAPl'ER II - TEE CAUCIIT PROBLEM FOR HIGEER ORDER EQUATIONS 1. Analytie funetions of several real variables • Formulation of the Cauehy problem. The not ion 2. of eharaeteristies. • • • 54 3. The Cauehy problem for the general non-linear equation. 71 4. The Cauehy-Kowalewsky theorem. 76 CHAPl'ER 111 - SECOND ORDER EQUATIONS WITH CONSTANT COEFFICIENTS 1. Equations in two independent variables. |
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analytic function apply arbitrary boundary data boundary value problem Cauchy data Cauchy problem characteristic curves characteristic equations characteristic strips coefficients completes the proof const converge defined directional derivative Dirichlet problem domain dxdy e)de easily verified equa existence follows formula fundamental solution given Green's function Green's identity harmonic function hence homogeneous equation hyperbolic initial conditions initial curve initial data initial manifold initial surface initial value problem integral equation integral surface Lemma linear equation mean value property neighborhood non-characteristic normal derivative numbers obtain order equation ordinary differential equations parameter family partial differential equation plane power series quantity quasi-linear Riemann function satisfy second order derivatives solve Suppose tangent theorem tion transformation uniformly unique vanish wave equation αξαη Σα ах ди ху ᎩᎩ