偏微分方程

內(nèi)容概要

　　《偏微分方程(第2版)》是一部講述偏微分方程理論的入門書籍。全書以橢圓偏微分為核心，系統(tǒng)講述了相關(guān)內(nèi)容，涉及到不少非線性問(wèn)題，如，最大值原理方法，拋物方程和變分法。書中講述了橢圓方程解的估計(jì)的主要方法，sobolev空間理論，弱解和強(qiáng)解，schauder估計(jì)，moser迭代。展示了橢圓，拋物和雙曲解以及布朗運(yùn)動(dòng)，半群之間的關(guān)系。《偏微分方程(第2版)》可以作為一年級(jí)的教程，在這新的版本中增加了反應(yīng)-擴(kuò)散方程和系統(tǒng)，新材料有neumann邊值問(wèn)題，poincaré不等式，以及一個(gè)新的證明，poisson方程解的hlder規(guī)則等。目次：以拉普拉斯方程為原型的二階橢圓偏微分方程；最大值原理；存在性技巧?。夯谧畲笾翟淼姆椒?；存在性技巧ⅱ：拋物方法.熱方程；反應(yīng)-擴(kuò)散方程和系統(tǒng)；波方程以及與laplace的關(guān)系和熱方程；熱方程，半群和布朗運(yùn)動(dòng)；dirichlet原理，pde解的變分法；sobolev空間和l2規(guī)范性理論；強(qiáng)解；schauder規(guī)范理論和連續(xù)性方法；moser迭代法和de
giorgi和nash規(guī)范性定理。
讀者對(duì)象：數(shù)學(xué)專業(yè)高年級(jí)的本科生，研究生和相關(guān)科研人員。

書籍目錄

introduction: what are partial differential equations?
1. the laplace equation as the prototype of an elliptic partial
differential equation of second order
1.1 harmonic functions. representation formula for the solution of
the dirichlet problem on the ball (existence techniques 0)
1.2 mean value properties of harmonic functions. subharmonic
functions. the maximum principle
2. the maximum principle
2.1 the maximum principle of e. hopf
2.2 the maximum principle of alexandrov and bakelman
2.3 maximum principles for nonlinear differential equations
3. existence techniques i: methods based on the maximum
principle
3.1 difference methods: discretization of differential
equations
3.2 the perron method
3.3 the alternating method of h.a. schwarz
3.4 boundary regularity
4. existence techniques ii: parabolic methods. the heat
equation
4.1 the heat equation: definition and maximum principles
4.2 the fundamental solution of the heat equation. the heat
equation and the laplace equation
4.3 the initial boundary value problem for the heat equation
4.4 discrete methods
5. reaction-diffusion equations and systems
5.1 reaction-diffusion equations
5.2 reaction-diffusion systems
5.3 the turing mechanism
6. the wave equation and its connections with the laplace and heat
equations
6.1 the one-dimensional wave equation
6.2 the mean value method: solving the wave equation through the
darboux equation
6.3 the energy inequality and the relation with the heat
equation
7. the heat equation, semigroups, and brownian motion
7.1 semigroups
7.2 infinitesimal generators of semigroups
7.3 brownian motion
8. the dirichlet principle. variational methods for the solu- tion
of pdes (existence techniques iii)
8.1 dirichlet's principle
8.2 the sobolev space w1,2
8.3 weak solutions of the poisson equation
8.4 quadratic variational problems
8.5 abstract hilbert space formulation of the variational prob-
lem. the finite element method
8.6 convex variational problems
9. sobolev spaces and l2 regularity theory
9.1 general sobolev spaces. embedding theorems of sobolev, morrey,
and john-nirenberg
9.2 l2-regularity theory: interior regularity of weak solutions of
the poisson equation
9.3 boundary regularity and regularity results for solutions of
general linear elliptic equations
9.4 extensions of sobolev functions and natural boundary con-
ditions
9.5 eigenvalues of elliptic operators
10. strong solutions
10.1 the regularity theory for strong solutions
10.2 a survey of the lp-regularity theory and applications to
solutions of semilinear elliptic equations
11. the regularity theory of schauder and the continuity method
(existence techniques iv)
11.1 ca-regularity theory for the poisson equation
11.2 the schauder estimates
11.3 existence techniques iv: the continuity method
12. the moser iteration method and the regularity theorem of de
giorgi and nash
12.1 the moser-harnack inequality
12.2 properties of solutions of elliptic equations
12.3 regularity of minimizers of variational problems
appendix. banach and hilbert spaces. the lp-spaces
references
index of notation
index

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