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 98
... ball contained in N and hence real analytic in the open ball . Thus harmonic functions are real analytic in the interior of their domain of definition . We can conclude from this that the Cauchy problem for the Laplace equation ...
... ball contained in N and hence real analytic in the open ball . Thus harmonic functions are real analytic in the interior of their domain of definition . We can conclude from this that the Cauchy problem for the Laplace equation ...
Página 112
... ball B ( § , p ) by the harmonic function that agrees with u on the boundary of the ball . The existence and uniqueness of u has been established earlier . ) Lemma II . For uЄo ( N ) and B ( § , p ) we have u ( x ) < u ( x ) for all x E ...
... ball B ( § , p ) by the harmonic function that agrees with u on the boundary of the ball . The existence and uniqueness of u has been established earlier . ) Lemma II . For uЄo ( N ) and B ( § , p ) we have u ( x ) < u ( x ) for all x E ...
Página 203
... ball y < 1 onto a neighborhood w of x1 and the half - ball y < 1 , Yn > 0 onto on . We choose for ( ( x ) a function in Co ( R " ) whose support is mapped into the ball { y || y | < } . Let Σ be the half - ball Σ = { ylye R " ; [ y ] ...
... ball y < 1 onto a neighborhood w of x1 and the half - ball y < 1 , Yn > 0 onto on . We choose for ( ( x ) a function in Co ( R " ) whose support is mapped into the ball { y || y | < } . Let Σ be the half - ball Σ = { ylye R " ; [ y ] ...
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