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 24
... definite way described by the equations - Fp pF2 + qFq dx dz dy dz Fa pF , + qFq ( 7.21a ) dp Fx + pFz dq = = dz pF2 + qFq dz F , + qF2 PF , + qFq ( 7.21b ) We can start out with an initial line element ( xo , Yo - Po , q % ) for z = z ...
... definite way described by the equations - Fp pF2 + qFq dx dz dy dz Fa pF , + qFq ( 7.21a ) dp Fx + pFz dq = = dz pF2 + qFq dz F , + qF2 PF , + qFq ( 7.21b ) We can start out with an initial line element ( xo , Yo - Po , q % ) for z = z ...
Página 140
... definite quadratic form in the first derivatives of u , we conclude that u , u = 0 for xЄ , t > 0 , hence that u is constant , and thus u = 0 because u = 0 initially . Consequently ( subject to appropriate regularity assumptions ) a ...
... definite quadratic form in the first derivatives of u , we conclude that u , u = 0 for xЄ , t > 0 , hence that u is constant , and thus u = 0 because u = 0 initially . Consequently ( subject to appropriate regularity assumptions ) a ...
Página 171
... definite for all ( x , t ) on S1 . Let A , A1 , ... , A " be symmetric , A positive definite , and R1 be space - like . Let u e C1 ( R ) be a solution of Set Lu = W for ( x , t ) = R2 ( 3.27d ) u = f ( x ) for t = 0 , | x | ≤ a ...
... definite for all ( x , t ) on S1 . Let A , A1 , ... , A " be symmetric , A positive definite , and R1 be space - like . Let u e C1 ( R ) be a solution of Set Lu = W for ( x , t ) = R2 ( 3.27d ) u = f ( x ) for t = 0 , | x | ≤ a ...
Contenido
Chapter | 1 |
Quasilinear Equations | 9 |
The Cauchy Problem | 24 |
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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