Mountain pass theorem
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The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.


Statement

The assumptions of the theorem are: * I is a functional from a Hilbert space ''H'' to the reals, * I\in C^1(H,\mathbb) and I' is
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
on bounded subsets of ''H'', * I satisfies the
Palais–Smale compactness condition The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular ...
, * I 0, * there exist positive constants ''r'' and ''a'' such that I geq a if \Vert u\Vert =r, and * there exists v\in H with \Vert v\Vert >r such that I leq 0. If we define: :\Gamma=\ and: :c=\inf_\max_ I mathbf(t) then the conclusion of the theorem is that ''c'' is a critical value of ''I''.


Visualization

The intuition behind the theorem is in the name "mountain pass." Consider ''I'' as describing elevation. Then we know two low spots in the landscape: the origin because I 0, and a far-off spot ''v'' where I leq 0. In between the two lies a range of mountains (at \Vert u\Vert =r) where the elevation is high (higher than ''a''>0). In order to travel along a path ''g'' from the origin to ''v'', we must pass over the mountains—that is, we must go up and then down. Since ''I'' is somewhat smooth, there must be a critical point somewhere in between. (Think along the lines of the mean-value theorem.) The mountain pass lies along the path that passes at the lowest elevation through the mountains. Note that this mountain pass is almost always a saddle point. For a proof, see section 8.5 of Evans.


Weaker formulation

Let X be Banach space. The assumptions of the theorem are: * \Phi\in C(X,\mathbf R) and have a
Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
\Phi'\colon X\to X^* which is continuous when X and X^* are endowed with
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the to ...
and
weak* topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
respectively. * There exists r>0 such that one can find certain \, x'\, >r with :\max\,(\Phi(0),\Phi(x'))<\inf\limits_\Phi(x)=:m(r). * \Phi satisfies weak Palais–Smale condition on \. In this case there is a critical point \overline x\in X of \Phi satisfying m(r)\le\Phi(\overline x). Moreover, if we define :\Gamma=\ then :\Phi(\overline x)=\inf_\max_\Phi(c\,(t)). For a proof, see section 5.5 of Aubin and Ekeland.


References


Further reading

* * * * * * {{cite book , first=Robert C. , last=McOwen , title=Partial Differential Equations: Methods and Applications , location=Upper Saddle River, NJ , publisher=Prentice Hall , year=1996 , isbn=0-13-121880-8 , pages=206–208 , chapter=Mountain Passes and Saddle Points , chapter-url=https://www.google.com/books/edition/_/TuNHsNC1Yf0C?hl=en&gbpv=1&pg=PA206 Mathematical analysis Calculus of variations Theorems in analysis