In the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of . It is a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

associated to the

Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet pr ...

for the semilinear

elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...

-\triangle u  = , u, ^u,\textu\mid_ = 0.

Here Δ is the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

on a bounded domain Ω in R^''n''. There are infinitely many solutions to this problem. Solutions are precisely the critical points for the

energy functional The energy functional is the total energy of a certain system, as a functional of the system's state. In the energy methods of simulating the dynamics of complex structures, a state of the system is often described as an element of an appropriat ...

J(v) = \frac12\int_-\frac1\int_

on the

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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...

. The Nehari manifold is defined to be the set of such that :

\, \nabla v\, ^2_ = \, v\, ^_ > 0.

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 :

\langle J'(u), u\rangle = 0.

Here ''J''′ is the

functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...

of ''J''.

References

* A. Bahri and P. L. Lions (1988), Morse Index of Some Min-Max Critical Points. I. Applications to Multiplicity Results. Communications on Pure and Applied Mathematics. (XLI) 1027–1037. * * * Calculus of variations {{mathanalysis-stub