偏微分方程（第2卷）

內(nèi)容概要

　　《偏微分方程(第2卷)》是一部兩卷集的偏微分方程教材。多變量橢圓，拋物和雙曲方程是研究的主要對象，解決了pde和多變量方法之間的關(guān)系。本書是第二卷主要講述了banach空間算子方程的可解性，hilbert空間線性算子和譜理論；線性橢圓微分方程的schauder理論；微分方程弱解；非線性偏微分方程；非線性橢圓系統(tǒng)和微分幾何應(yīng)用。書中各章的獨(dú)立性較強(qiáng)，有一定偏微分方程基本知識的讀者可以獨(dú)立閱讀各章。目次：banach空間中的算子；hilbert空間線性算子；線性橢圓微分方程；非線性偏微分方程；非線性橢圓系統(tǒng)。
　　讀者對象：數(shù)學(xué)專業(yè)的本科生，研究生和相關(guān)的科研人員。

書籍目錄

vii operators in banach spaces
　1 fixed point theorems
　2 the leray-schauder degree of mapping
　3 fundamental properties for the degree of mapping
　4 linear operators in banach spaces
　5 some historical notices to the chapters iii and vii
viii linear operators in hilbert spaces
　1 various eigenvalue problems
　2 singular integral equations
　3 the abstract hilbert space
　4 bounded linear operators in hilbert spaces
　5 unitary operators
　6 completely continuous operators in hilbert spaces
　7 spectral theory for completely continuous hermitian
operators
　8 the sturm-liouville eigenvalue problem
　9 weyl's eigenvalue problem for the laplace operator
　9 some historical notices to chapter viii
ix linear elliptic differential equations
　1 the differential equation △φ+p(x, y)φx+q(x, y)φy=r(x, y)
　2 the schwarzian integral formula
　3 the riemann-hilbert boundary value problem
　4 potential-theoretic estimates.
　5 schauder's continuity method
　6 existence and regularity theorems
　7 the schauder estimates
　8 some historical notices to chapter ix
x weak solutions of elliptic differential equations
　1 sobolev spaces
　2 embedding and compactness
　3 existence of weak solutions
　4 boundedness of weak solutions
　5 hslder continuity of weak solutions
　6 weak potential-theoretic estimates
　7 boundary behavior of weak solutions
　8 equations in divergence form
　9 green's function for elliptic operators
　10 spectral theory of the laplace-beltrami operator
　11 some historical notices to chapter x
xi nonlinear partial differential equations
　1 the fundamental forms and curvatures of a surface
　2 two-dimensional parametric integrals
　3 quasilinear hyperbolic differential equations and systems of
second order (characteristic parameters)
　4 cauchy's initial value problem for quasilinear hyperbolic
differential equations and systems of second order
　5 riemann's integration method
　6 bernstein's analyticity theorem
　7 some historical notices to chapter xi
xii nonlinear elliptic systems
　1 maximum principles for the h-surface system
　2 gradient estimates for nonlinear elliptic systems
　3 global estimates for nonlinear systems
　4 the dirichlet problem for nonlinear elliptic systems
　5 distortion estimates for plane elliptic systems
　6 a curvature estimate for minimal surfaces
　7 global estimates for conformal mappings with respect to
riemannian metrics
　8 introduction of conformal parameters into a riemannian
metric
　9 the uniformization method for quasilinear elliptic differential
equations and the dirichlet problem
　10 an outlook on plateau's problem
　11 some historical notices to chapter xii
references
index

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