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 106
... assumption that u is bounded from above in N. ( The additional assumption is needed to exclude examples like u = x2 . ) [ Hint : Take for & > 0 the harmonic function u ( x1 , x2 ) − e log√x } + ( x2 + 1 ) 2 . Apply the maximum ...
... assumption that u is bounded from above in N. ( The additional assumption is needed to exclude examples like u = x2 . ) [ Hint : Take for & > 0 the harmonic function u ( x1 , x2 ) − e log√x } + ( x2 + 1 ) 2 . Apply the maximum ...
Página 226
... assumptions of the uniqueness theorem on p . 216. It follows that W ( x , s ) = 0 for x = R , 0 ≤ s ≤ T − 2ɛ , and ... assumption ( 1.57c ) is replaced by u ( x , t ) > -c for xe R , 0 < t < T with any constant c , or by u ( x 226 7 ...
... assumptions of the uniqueness theorem on p . 216. It follows that W ( x , s ) = 0 for x = R , 0 ≤ s ≤ T − 2ɛ , and ... assumption ( 1.57c ) is replaced by u ( x , t ) > -c for xe R , 0 < t < T with any constant c , or by u ( x 226 7 ...
Página 227
... assumption that the coefficients a , b , c belong to C ( ) and that they and each of their partial derivatives are bounded uniformly in 2. ( Actually only a small finite number of deriva- tives will be required in each theorem ) . Most ...
... assumption that the coefficients a , b , c belong to C ( ) and that they and each of their partial derivatives are bounded uniformly in 2. ( Actually only a small finite number of deriva- tives will be required in each theorem ) . Most ...
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