Polyharmonic boundary value problems : positivity preserving and nonlinear higher order elliptic equations in bounded domains
Author(s)
Bibliographic Information
Polyharmonic boundary value problems : positivity preserving and nonlinear higher order elliptic equations in bounded domains
（Lecture notes in mathematics, 1991）
Springer, c2010
- : pbk
Available at / 57 libraries
-
Library, Research Institute for Mathematical Sciences, Kyoto University数研
: pbkL/N||LNM||1991200017822733
-
No Libraries matched.
- Remove all filters.
Note
Bibliography: p. 397-413
Includes indexes
Description and Table of Contents
Description
Linear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat eq- tion or,equivalently,the membraneequation. Forthis intensivelywell-studiedlinear problem there are two main lines of results. The ?rst line consists of existence and regularity results. Usually the solution exists and "gains two orders of differen- ation" with respect to the source term. The second line contains comparison type results, namely the property that a positive source term implies that the solution is positive under suitable side constraints such as homogeneous Dirichlet bou- ary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of results hold for general second order elliptic problems, see the books by Gilbarg-Trudinger [198] and Protter-Weinberger [347]. For linear higher order elliptic problems the existence and regularitytype results - main, as one may say, in their full generality whereas comparison type results may fail. Here and in the sequel "higher order" means order at least four. Most interesting models, however, are nonlinear.
By now, the theory of second order elliptic problems is quite well developed for semilinear, quasilinear and even for some fully nonlinear problems. If one looks closely at the tools being used in the proofs, then one ?nds that many results bene?t in some way from the positivity preserving property. Techniques based on Harnack's inequality, De Giorgi-Nash- Moser's iteration, viscosity solutions etc.
Table of Contents
Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.- Willmore Surfaces of Revolution.
by "Nielsen BookData"
