Linking Systems

Martin Schechter

Chapter Chapter 1 in Critical Point Theory, 2020, pp 1-20 from Springer

Abstract: Abstract Many problems arising in science and engineering call for the solving of the Euler equations of functionals, i.e., equations of the form Euler equations G ′ ( u ) = 0 , $$\displaystyle G'(u)=0, $$ where Functional G(u) is a C 1 functional Energy (usually representing the energy) arising from the given data. As an illustration, the equation − Δ u ( x ) = f ( x , u ( x ) ) $$\displaystyle -\Delta u(x)=f(x,u(x)) $$ is the Euler equation of the functional G ( u ) = 1 2 ∥ ∇ u ∥ 2 − ∫ F ( x , u ( x ) ) d x $$\displaystyle G(u)=\frac {1}{2}\|\nabla u\|{ }^2-\int F(x,u(x)) \,dx $$ on an appropriate space, where F ( x , t ) = ∫ 0 t f ( x , s ) d s , $$\displaystyle F(x,t) = \int ^{t}_{0}f(x,s)\,ds, $$ and the norm is that of L 2. The solving of the Euler equations is tantamount to finding critical points of the corresponding functional. The history of this approach goes back to the calculus of variations. Calculus of variations Then the desire was to find extrema of certain expressions G (functionals). Following the approach of calculus, one tried to find all critical points of G, substitute them back in G, and see which one gives the required extremum. Critical points This worked fairly well in one dimension where G′(u) = 0 is an ordinary differential equation. However, in higher dimensions, it turned out that it was easier to find the extrema of G than solve G′(u) = 0. This led to the approach of solving equations of the form G′(u) = 0 by finding extrema of G.

Date: 2020
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-45603-0_1

Ordering information: This item can be ordered from
http://www.springer.com/9783030456030

DOI: 10.1007/978-3-030-45603-0_1

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().