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 88
... Take the wave equation for u = u ( x1 , x2 , X3 ) . Let Z be the disc ( 5.7a ) Lu = Ux3x3 - Ux1x1 - Ux2x2 0 = x2 + x3 < 1 ; x1 = 0 ( 5.7b ) = in the " time - like ' plane x1 = 0. Show that every cone with base Z and vertex y = ( 1,2,3 ) ...
... Take the wave equation for u = u ( x1 , x2 , X3 ) . Let Z be the disc ( 5.7a ) Lu = Ux3x3 - Ux1x1 - Ux2x2 0 = x2 + x3 < 1 ; x1 = 0 ( 5.7b ) = in the " time - like ' plane x1 = 0. Show that every cone with base Z and vertex y = ( 1,2,3 ) ...
Página 122
... Take any z EN and p so small that the ball B ( z , 3p ) lies in N. Take any test function with support in B ( z , p ) and a fixed test function ( x ) which has the value 1 in B ( z , 2p ) . Set u ( x ) = √ K ( x , § ) o ( § ) d ...
... Take any z EN and p so small that the ball B ( z , 3p ) lies in N. Take any test function with support in B ( z , p ) and a fixed test function ( x ) which has the value 1 in B ( z , 2p ) . Set u ( x ) = √ K ( x , § ) o ( § ) d ...
Página 150
... Take for each k = 1 , ... , m the open disk of center and radius 1 in the complex λ - plane . Let U denote the union of these m ( possibly overlapping ) disks . Take for the path of integration г in ( 2.31 ) the boundary of U , which ...
... Take for each k = 1 , ... , m the open disk of center and radius 1 in the complex λ - plane . Let U denote the union of these m ( possibly overlapping ) disks . Take for the path of integration г in ( 2.31 ) the boundary of U , which ...
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