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. |
Dentro del libro
Resultados 1-3 de 51
Página 105
... Apply maximum principle to spherical shell . ] 2. Let u ( x ) = C2 ( N ) Co ( M ) be a solution of n 2 Au + Σ ax ( x ) uxx + c ( x ) u = 0 , k = 1 ( 2.6 ) where c ( x ) < 0 in 2. Show that u = 0 on N implies u = 0 in N. [ Hint : Show ...
... Apply maximum principle to spherical shell . ] 2. Let u ( x ) = C2 ( N ) Co ( M ) be a solution of n 2 Au + Σ ax ( x ) uxx + c ( x ) u = 0 , k = 1 ( 2.6 ) where c ( x ) < 0 in 2. Show that u = 0 on N implies u = 0 in N. [ Hint : Show ...
Página 106
... Apply the maximum principle to a region x2 + ( x2 + 1 ) 2 < a2 , x2 > 0 with large a . Let ε → 0 . ] 3. The Dirichlet Problem , Green's Function , and Poisson's Formula We derived the representation ( 1.23 ) for u in terms of Cauchy ...
... Apply the maximum principle to a region x2 + ( x2 + 1 ) 2 < a2 , x2 > 0 with large a . Let ε → 0 . ] 3. The Dirichlet Problem , Green's Function , and Poisson's Formula We derived the representation ( 1.23 ) for u in terms of Cauchy ...
Página 109
Fritz John. As an application we derive estimates for the derivatives of a harmonic function u in terms of estimates ... apply ( 3.13b ) to a ball of any radius a < d ( § ) with center at § . For a → d ( § ) we obtain the inequality n ...
Fritz John. As an application we derive estimates for the derivatives of a harmonic function u in terms of estimates ... apply ( 3.13b ) to a ball of any radius a < d ( § ) with center at § . For a → d ( § ) we obtain the inequality n ...
Contenido
Chapter | 1 |
Examples | 2 |
Analytic Solution and Approximation Methods in a Simple Example Problems 4 Quasilinear Equations | 4 |
Derechos de autor | |
Otras 35 secciones no mostradas
Otras ediciones - Ver todas
Términos y frases comunes
analytic functions assume ball boundary bounded uniformly Cauchy data Cauchy problem Cauchy sequence Chapter characteristic curves class C² coefficients compact support complex constant continuous converge defined denote derivatives of orders difference quotients Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f fundamental solution Gårding Gårding's inequality given harmonic function heat equation hence Hint Ho(N holds identity implies inequality initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix maximum principle neighborhood non-characteristic norm obtained open set partial differential equation plane polynomial power series prescribed proof real analytic real numbers satisfies scalar Show solution u(x,t solved space square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ΘΩ