Nehari manifold
Encyclopedia
In the calculus of variations
, a branch of mathematics
, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differential manifold associated to the Dirichlet problem
for the semilinear elliptic partial differential equation
Here Δ is the Laplacian on a bounded domain Ω in Rn.
There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional
on the Sobolev space
. The Nehari manifold is defined to be the set of such that
Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear.
More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which
Here J′ is the functional derivative of J.
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a differential manifold associated to the Dirichlet problem
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
for the semilinear elliptic partial differential equation
Here Δ is the Laplacian on a bounded domain Ω in Rn.
There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional
on the Sobolev space
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
. The Nehari manifold is defined to be the set of such that
Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear.
More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which
Here J′ is the functional derivative of J.