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 102
... Prove that △ u ( x ) = Au ( x1 , xn ) = 0 also implies that - A ( x2 = " u ( x / | x | 2 ) ) = 0 ( 1.38 ) for x / x2 in the domain of definition of u . 6. Let x , y denote coordinates in the plane . Let u ( x , y ) be a solution of the ...
... Prove that △ u ( x ) = Au ( x1 , xn ) = 0 also implies that - A ( x2 = " u ( x / | x | 2 ) ) = 0 ( 1.38 ) for x / x2 in the domain of definition of u . 6. Let x , y denote coordinates in the plane . Let u ( x , y ) be a solution of the ...
Página 103
Fritz John. ( a ) Prove that F - gradu , where the " potential " u is given by μ ( χ ) u ( 5 ) = Y - ~ S S S μ ( x ) dx . - ( b ) Prove that the attraction F ( § ) exerted by 2 on a far away unit mass is approximately the same as if the ...
Fritz John. ( a ) Prove that F - gradu , where the " potential " u is given by μ ( χ ) u ( 5 ) = Y - ~ S S S μ ( x ) dx . - ( b ) Prove that the attraction F ( § ) exerted by 2 on a far away unit mass is approximately the same as if the ...
Página 125
... prove that the w * form a Cauchy sequence converging to an element w of H with ( w ) = 1 , wm . Show that ( w , u ) = 0 for ( u ) = 0 , and finally that ( v , u ) = 4 ( u ) for v = w / ( w , w ) and all u . ] 2. Show that a solution ...
... prove that the w * form a Cauchy sequence converging to an element w of H with ( w ) = 1 , wm . Show that ( w , u ) = 0 for ( u ) = 0 , and finally that ( v , u ) = 4 ( u ) for v = w / ( w , w ) and all u . ] 2. Show that a solution ...
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