Partial Differential EquationsSpringer, 1982 - 249 páginas This book is a very well-accepted introduction to the subject. In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis. Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy 's example of a linear equation without solutions. |
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Página 213
... uniformly with its derivatives of order < m in any bounded set containing the support of u . 2. Let f ( x ) have uniformly bounded derivatives of orders < s . Show that the u ( x , t ) given by ( 1.11 ) is of class C for t≥ 0 and all x ...
... uniformly with its derivatives of order < m in any bounded set containing the support of u . 2. Let f ( x ) have uniformly bounded derivatives of orders < s . Show that the u ( x , t ) given by ( 1.11 ) is of class C for t≥ 0 and all x ...
Página 230
... uniformly bounded domain of dependence for v ' ( x , t ) on ƒ , and hence also for u ( x , t ) , whereas the example of the heat equation shows that the domain of dependence of u ( x , t ) on the initial values is the whole x - axis ...
... uniformly bounded domain of dependence for v ' ( x , t ) on ƒ , and hence also for u ( x , t ) , whereas the example of the heat equation shows that the domain of dependence of u ( x , t ) on the initial values is the whole x - axis ...
Página 232
... uniformly in Σ ,, and hence also in Σ for μv . For a suitable subsequence of the integers v lim v ' ( x , t ) = u ... uniformly Lipschitz in Σ ,, the limits u , u ' , u " , ù are uniformly Lipschitz in U , and hence can be extended as ...
... uniformly in Σ ,, and hence also in Σ for μv . For a suitable subsequence of the integers v lim v ' ( x , t ) = u ... uniformly Lipschitz in Σ ,, the limits u , u ' , u " , ù are uniformly Lipschitz in U , and hence can be extended as ...
Contenido
Chapter | 1 |
Examples | 2 |
Analytic Solution and Approximation Methods in a Simple Example Problems 4 Quasilinear Equations | 4 |
Derechos de autor | |
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analytic functions assume ball boundary bounded uniformly Cauchy data Cauchy problem Cauchy sequence Chapter characteristic curves class C² coefficients compact support complex constant continuous converge defined denote derivatives of orders difference quotients Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f fundamental solution Gårding Gårding's inequality given harmonic function heat equation hence Hint Ho(N holds identity implies inequality initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix maximum principle neighborhood non-characteristic norm obtained open set partial differential equation plane polynomial power series prescribed proof real analytic real numbers satisfies scalar Show solution u(x,t solved space square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ΘΩ