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 7 . Now this is true for ( B ( § , p ) by ( 4.4a ) , since Max Prin ( 4.7 ) mutu ( 8 ) 2 int Suads 1x - 21 = 0 uo ( ) , and is true for ( E B ( § , p ) since then u ,, is harmonic near § . There remains the case where ...
... sufficiently small 7 . Now this is true for ( B ( § , p ) by ( 4.4a ) , since Max Prin ( 4.7 ) mutu ( 8 ) 2 int Suads 1x - 21 = 0 uo ( ) , and is true for ( E B ( § , p ) since then u ,, is harmonic near § . There remains the case where ...
Página 168
... sufficiently small . Hence for sufficiently small T we have гMn < 1/2 , and conclude from ( 3.20d ) that Σ || D , u || ≤ 2nMT ( 1 + T ) || w || + 25 Σ || D ; w || . ( 3.21 ) A similar estimate can then be obtained for || Tu || directly ...
... sufficiently small . Hence for sufficiently small T we have гMn < 1/2 , and conclude from ( 3.20d ) that Σ || D , u || ≤ 2nMT ( 1 + T ) || w || + 25 Σ || D ; w || . ( 3.21 ) A similar estimate can then be obtained for || Tu || directly ...
Página 181
... sufficiently small , under the assumption that wЄ Cn + 2 ( R ) 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 ( R ) 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 |
Quasilinear Equations | 9 |
The Cauchy Problem | 24 |
Derechos de autor | |
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assume ball boundary bounded uniformly Cauchy data Cauchy problem Cauchy sequence Cauchy-Kowalevski Chapter characteristic curves class C² coefficients compact support complex constant continuous converge defined denote derivatives of orders determined uniquely difference quotients Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f(x fundamental solution Gårding given Green's identity heat equation hence Hint holds identity implies inequality initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix neighborhood non-characteristic norm obtained open set partial differential equation plane polynomial power series prescribed proof quasi-linear real numbers satisfies scalar second derivatives Show solved space square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ΘΩ