An Introduction To Maximum Principles And Symmetry In Elliptic Problems (cambridge Tracts In Mathematics)
by L. E. Fraenkel /
2000 / English / PDF
5.2 MB Download
This book presents the basic theory of the symmetry of solutions to
second-order elliptic partial differential equations by means of
the maximum principle. It proceeds from elementary facts about the
linear case to recent results about positive solutions of nonlinear
elliptic equations. Gidas, Ni and Nirenberg, building on the work
of Alexandrov and Serrin, have shown that the shape of the set on
which such elliptic equations are solved has a strong effect on the
form of positive solutions. In particular, if the equation and its
boundary condition allow spherically symmetric solutions, then,
remarkably, all positive solutions are spherically symmetric. These
recent and important results are presented with minimal
prerequisites, in a style suited to graduate students. Two long
appendices give a leisurely account of basic facts about the
Laplace and Poisson equations, and there is an abundance of
exercises, with detailed hints, some of which contain new results.
This book presents the basic theory of the symmetry of solutions to
second-order elliptic partial differential equations by means of
the maximum principle. It proceeds from elementary facts about the
linear case to recent results about positive solutions of nonlinear
elliptic equations. Gidas, Ni and Nirenberg, building on the work
of Alexandrov and Serrin, have shown that the shape of the set on
which such elliptic equations are solved has a strong effect on the
form of positive solutions. In particular, if the equation and its
boundary condition allow spherically symmetric solutions, then,
remarkably, all positive solutions are spherically symmetric. These
recent and important results are presented with minimal
prerequisites, in a style suited to graduate students. Two long
appendices give a leisurely account of basic facts about the
Laplace and Poisson equations, and there is an abundance of
exercises, with detailed hints, some of which contain new results.