Some asymptotic problems in the theory of partial differential equations. (English) Zbl 1075.35500

Lezioni Lincee. Cambridge: Cambridge Univ. Press (ISBN 0-521-48537-1/pbk; 0-521-48083-3/hbk). x, 202 p. (1996).
This book contains results obtained by the author in recent years. Most of them are published here for the first time. About one half of the book is devoted to the asymptotic behaviour, at infinity, of solutions to nonlinear elliptic second-order equations, in the cases of both Dirichlet and Neumann boundary conditions. First, in Chapter 1, the case of general unbounded domains is studied. Next, in Chapter 2, special attention is paid to the case of cylindrical domains. The author considers equations with and without first-order terms, as well as particular equations \(\Delta u - |u|^{p-1}u =0\) and \(\Delta u - e^u=0\), these cases being important in various applications.
Chapter 3 deals with the asymptotic behaviour of solutions near a conic point of the boundary. Again, both Dirichlet and Neumann conditions are considered. Finally, in Chapter 4, the author investigates the homogenization problem for linear elliptic equations in partially perforated domains. This means that the underlying domain is divided into two parts; the first one is perforated, while the second contains no holes.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations