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 113
... sufficiently small T. Now this is true for ( 4.6 ) B ( § , p ) by ( 4.4a ) , since uẸσ ( N ) , and is true for 3 E B ( § , p ) since then u ,, is harmonic near § . There remains the case where & ES ( § , p ) . But then by ( 4.2 ) ...
... sufficiently small T. Now this is true for ( 4.6 ) B ( § , p ) by ( 4.4a ) , since uẸσ ( N ) , and is true for 3 E B ( § , p ) since then u ,, is harmonic near § . There remains the case where & ES ( § , p ) . But then by ( 4.2 ) ...
Página 168
... sufficiently small . Hence for sufficiently small T we have гMn < 1/2 , and conclude from ( 3.20d ) that ΣD ; u || ≤ 2nMг ( 1 + г ) || w || + 2г Σ || D ; w || . 2Γ 25 i ( 3.21 ) A similar estimate can then be obtained for || ru ...
... sufficiently small . Hence for sufficiently small T we have гMn < 1/2 , and conclude from ( 3.20d ) that ΣD ; u || ≤ 2nMг ( 1 + г ) || w || + 2г Σ || D ; w || . 2Γ 25 i ( 3.21 ) A similar estimate can then be obtained for || ru ...
Página 181
... sufficiently small , under the assumption that wЄ Cn + 2 ( RT ) and that w has compact support with respect to x . * PROBLEMS 1. Show that for the solution u of ( 3.1a , b ) constructed here the domain of dependence of u ( x , t ) on w ...
... sufficiently small , under the assumption that wЄ Cn + 2 ( RT ) and that w has compact support with respect to x . * PROBLEMS 1. Show that for the solution u of ( 3.1a , b ) constructed here the domain of dependence of u ( x , t ) on w ...
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