Elliptic & Parabolic Equations
This book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between these two kinds of equations as well as contrast the similarities and differences between them.
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L2 Theory of Linear Elliptic Equations
L2 Theory of Linear Parabolic Equations
De Giorgi Iteration and Moser Iteration
Schauders Estimates for Linear Elliptic Equations
Schauders Estimates for Linear Parabolic Equations
Existence of Classical Solutions for Linear Equations
Fixed Point Method
Topological Degree Method
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admits a unique Assume Banach space boundary value condition bounded domain bounded linear BR(x C(QT Cauchy's inequality Chapter choose classical solutions converges Corollary cut-off function defined definition of weak Denote derive Dirichlet problem dpQT Dw\2dxdt dxdt elliptic equations embedding theorem establish exists a constant G QT Harnack's inequality heat equation Hence Holder's estimate Hq(Q implies initial-boundary value problem integrating interior estimate JJq+ JJQp JJQr JJQt JJQt Jn Jn Laplace's equation Let u G Lipschitz continuous LP estimates LP norm mapping maximum principle nondecreasing ordered supersolution parabolic equations Poincare's inequality Poisson's equation positive constant depending Prom proof of Theorem Proposition quasilinear quasimonotone Remark satisfies Schauder's estimates sequence Sobolev spaces solution of equation solution of problem space strong solution supersolution and subsolution supu u2dx u2dxdt unique solution weak derivatives weak solution weak subsolution x,t)eQT