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 107
... given boundary values u = f on a formula ( 3.7 ) solves the Dirichlet problem , provided that problem has a solution uЄC2 ) . We shall verify directly that for ƒ continuous on a the problem actually has a solution given by Poisson's ...
... given boundary values u = f on a formula ( 3.7 ) solves the Dirichlet problem , provided that problem has a solution uЄC2 ) . We shall verify directly that for ƒ continuous on a the problem actually has a solution given by Poisson's ...
Página 160
... given by ( 2.31 ) . m- 1 u ( x , t ) = eix.¿Z ( ¿ , t ) im · ( 2.75b ) ( 2.75c ) We now restrict ourselves to the case that P is strictly hyperbolic , that is , that all roots of the homogeneous equation are real and distinct for $ 0 ...
... given by ( 2.31 ) . m- 1 u ( x , t ) = eix.¿Z ( ¿ , t ) im · ( 2.75b ) ( 2.75c ) We now restrict ourselves to the case that P is strictly hyperbolic , that is , that all roots of the homogeneous equation are real and distinct for $ 0 ...
Página 213
... given by ( 1.11 ) is of class Cs for t > 0 and all x . [ Hint : Show that Du = [ KDfdy . ] 3. Let f ( x ) be continuous in R " and satisfy ( 1.14 ) . Show that the u defined by ( 1.11 ) is analytic in x , t for all complex x and complex ...
... given by ( 1.11 ) is of class Cs for t > 0 and all x . [ Hint : Show that Du = [ KDfdy . ] 3. Let f ( x ) be continuous in R " and satisfy ( 1.14 ) . Show that the u defined by ( 1.11 ) is analytic in x , t for all complex x and complex ...
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