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 109
... inequality || ( § ) < √ ( 5 ) sup | u | . ( 3.13c ) Similar inequalities obviously hold for higher derivatives also . Let w denote a compact subset of . Then there exists a positive lower bound for d ( § ) in w . All harmonic u with a ...
... inequality || ( § ) < √ ( 5 ) sup | u | . ( 3.13c ) Similar inequalities obviously hold for higher derivatives also . Let w denote a compact subset of . Then there exists a positive lower bound for d ( § ) in w . All harmonic u with a ...
Página 111
... inequality ( 3.13b ) can be replaced by 2ηωη - 1 ( 3.16 ) = Yn - ( n − 1 ) wn ( 3.17 ) and that this is the smallest possible constant in the inequality . [ Hint : Use u = sgn x , on | x | = a in ( 3.13a ) . ] 9. Let Au ( x ) = 0 and ...
... inequality ( 3.13b ) can be replaced by 2ηωη - 1 ( 3.16 ) = Yn - ( n − 1 ) wn ( 3.17 ) and that this is the smallest possible constant in the inequality . [ Hint : Use u = sgn x , on | x | = a in ( 3.13a ) . ] 9. Let Au ( x ) = 0 and ...
Página 168
... inequalities . This inequality can easily be derived from Parseval's identity , known from the theory of Fourier integrals , that connects the square integrals of a function g with that of its Fourier transform ĝ : S \ 8 ( x ) | 2dx = f ...
... inequalities . This inequality can easily be derived from Parseval's identity , known from the theory of Fourier integrals , that connects the square integrals of a function g with that of its Fourier transform ĝ : S \ 8 ( x ) | 2dx = f ...
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