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 110
... vanish for xn 0. Extend u = as an odd function of x , to all of B. Show that the extended u is harmonic in B. [ Hint : Let u * be the harmonic function in B with boundary values u on dB . Then u * is odd in x ,, and u * = u in B + ...
... vanish for xn 0. Extend u = as an odd function of x , to all of B. Show that the extended u is harmonic in B. [ Hint : Let u * be the harmonic function in B with boundary values u on dB . Then u * is odd in x ,, and u * = u in B + ...
Página 119
... vanishing solutions u = const . Denote by C ( ) the subspace of functions u in C1 ( ) that vanish on . Obviously ( u , v ) can be used as inner product on C with the corresponding squared norm given by the Dirichlet integral || || 2 ...
... vanishing solutions u = const . Denote by C ( ) the subspace of functions u in C1 ( ) that vanish on . Obviously ( u , v ) can be used as inner product on C with the corresponding squared norm given by the Dirichlet integral || || 2 ...
Página 142
... vanish . = We observe that the consistency conditions ( 1.69 ) are satisfied automati- cally , when h = 0 and in addition f , g , w vanish for all sufficiently small x3 . When w = h = 0 and ( 1.69 ) holds , we can reduce the mixed ...
... vanish . = We observe that the consistency conditions ( 1.69 ) are satisfied automati- cally , when h = 0 and in addition f , g , w vanish for all sufficiently small x3 . When w = h = 0 and ( 1.69 ) holds , we can reduce the mixed ...
Contenido
Chapter | 1 |
Quasilinear Equations | 9 |
The Cauchy Problem | 24 |
Derechos de autor | |
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Términos y frases comunes
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