• Source: Mountain pass theorem

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


{\displaystyle I}

is a functional from a Hilbert space H to the reals,




I
∈

C

1


(
H
,

R

)


{\displaystyle I\in C^{1}(H,\mathbb {R} )}

and




I
′



{\displaystyle I'}

is Lipschitz continuous on bounded subsets of H,




I


{\displaystyle I}

satisfies the Palais–Smale compactness condition,




I
[
0
]
=
0


{\displaystyle I[0]=0}

,
there exist positive constants r and a such that



I
[
u
]
≥
a


{\displaystyle I[u]\geq a}

if



‖
u
‖
=
r


{\displaystyle \Vert u\Vert =r}

, and
there exists



v
∈
H


{\displaystyle v\in H}

with



‖
v
‖
>
r


{\displaystyle \Vert v\Vert >r}

such that



I
[
v
]
≤
0


{\displaystyle I[v]\leq 0}

.
If we define:




Γ
=
{

g

∈
C
(
[
0
,
1
]
;
H
)

|


g

(
0
)
=
0
,

g

(
1
)
=
v
}


{\displaystyle \Gamma =\{\mathbf {g} \in C([0,1];H)\,\vert \,\mathbf {g} (0)=0,\mathbf {g} (1)=v\}}


and:




c
=

inf


g

∈
Γ



max

0
≤
t
≤
1


I
[

g

(
t
)
]
,


{\displaystyle c=\inf _{\mathbf {g} \in \Gamma }\max _{0\leq t\leq 1}I[\mathbf {g} (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
]
=
0


{\displaystyle I[0]=0}

, and a far-off spot v where



I
[
v
]
≤
0


{\displaystyle I[v]\leq 0}

. In between the two lies a range of mountains (at



‖
u
‖
=
r


{\displaystyle \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


{\displaystyle X}

be Banach space. The assumptions of the theorem are:




Φ
∈
C
(
X
,

R

)


{\displaystyle \Phi \in C(X,\mathbf {R} )}

and have a Gateaux derivative




Φ
′

:
X
→

X

∗




{\displaystyle \Phi '\colon X\to X^{*}}

which is continuous when



X


{\displaystyle X}

and




X

∗




{\displaystyle X^{*}}

are endowed with strong topology and weak* topology respectively.
There exists



r
>
0


{\displaystyle r>0}

such that one can find certain



‖

x
′

‖
>
r


{\displaystyle \|x'\|>r}

with




max

(
Φ
(
0
)
,
Φ
(

x
′

)
)
<

inf

‖
x
‖
=
r


Φ
(
x
)
=:
m
(
r
)


{\displaystyle \max \,(\Phi (0),\Phi (x'))<\inf \limits _{\|x\|=r}\Phi (x)=:m(r)}

.




Φ


{\displaystyle \Phi }

satisfies weak Palais–Smale condition on



{
x
∈
X
∣
m
(
r
)
≤
Φ
(
x
)
}


{\displaystyle \{x\in X\mid m(r)\leq \Phi (x)\}}

.
In this case there is a critical point





x
¯


∈
X


{\displaystyle {\overline {x}}\in X}

of



Φ


{\displaystyle \Phi }

satisfying



m
(
r
)
≤
Φ
(


x
¯


)


{\displaystyle m(r)\leq \Phi ({\overline {x}})}

. Moreover, if we define




Γ
=
{
c
∈
C
(
[
0
,
1
]
,
X
)
∣
c

(
0
)
=
0
,

c

(
1
)
=

x
′

}


{\displaystyle \Gamma =\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x'\}}


then




Φ
(


x
¯


)
=

inf

c

∈

Γ



max

0
≤
t
≤
1


Φ
(
c

(
t
)
)
.


{\displaystyle \Phi ({\overline {x}})=\inf _{c\,\in \,\Gamma }\max _{0\leq t\leq 1}\Phi (c\,(t)).}


For a proof, see section 5.5 of Aubin and Ekeland.




References






Further reading


Aubin, Jean-Pierre; Ekeland, Ivar (2006). Applied Nonlinear Analysis. Dover Books. ISBN 0-486-45324-3.
Bisgard, James (2015). "Mountain Passes and Saddle Points". SIAM Review. 57 (2): 275–292. doi:10.1137/140963510.
Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2.
Jabri, Youssef (2003). The Mountain Pass Theorem, Variants, Generalizations and Some Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press. ISBN 0-521-82721-3.
Mawhin, Jean; Willem, Michel (1989). "The Mountain Pass Theorem and Periodic Solutions of Superlinear Convex Autonomous Hamiltonian Systems". Critical Point Theory and Hamiltonian Systems. New York: Springer-Verlag. pp. 92–97. ISBN 0-387-96908-X.
McOwen, Robert C. (1996). "Mountain Passes and Saddle Points". Partial Differential Equations: Methods and Applications. Upper Saddle River, NJ: Prentice Hall. pp. 206–208. ISBN 0-13-121880-8.

Kata Kunci Pencarian:

• Mountain pass theorem
• Path of least resistance
• Saddle point
• List of theorems
• Antonio Ambrosetti
• Morse theory
• Louis Nirenberg
• Palais–Smale compactness condition
• Paul Rabinowitz
• Likelihood function