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.[1][2] 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} {\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} )} {\displaystyle I\in C^{1}(H,\mathbb {R} )} and I ′ {\displaystyle I'} {\displaystyle I'} is Lipschitz continuous on bounded subsets of H,
  • I {\displaystyle I} {\displaystyle I} satisfies the Palais–Smale compactness condition,
  • I [ 0 ] = 0 {\displaystyle I[0]=0} {\displaystyle I[0]=0},
  • there exist positive constants r and a such that I [ u ] ≥ a {\displaystyle I[u]\geq a} {\displaystyle I[u]\geq a} if ‖ u ‖ = r {\displaystyle \Vert u\Vert =r} {\displaystyle \Vert u\Vert =r}, and
  • there exists v ∈ H {\displaystyle v\in H} {\displaystyle v\in H} with ‖ v ‖ > r {\displaystyle \Vert v\Vert >r} {\displaystyle \Vert v\Vert >r} such that I [ v ] ≤ 0 {\displaystyle I[v]\leq 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\}} {\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)],} {\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} {\displaystyle I[0]=0}, and a far-off spot v where I [ v ] ≤ 0 {\displaystyle I[v]\leq 0} {\displaystyle I[v]\leq 0}. In between the two lies a range of mountains (at ‖ u ‖ = r {\displaystyle \Vert u\Vert =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} {\displaystyle X} be Banach space. The assumptions of the theorem are:

  • Φ ∈ C ( X , R ) {\displaystyle \Phi \in C(X,\mathbf {R} )} {\displaystyle \Phi \in C(X,\mathbf {R} )} and have a Gateaux derivative Φ ′ : X → X ∗ {\displaystyle \Phi '\colon X\to X^{*}} {\displaystyle \Phi '\colon X\to X^{*}} which is continuous when X {\displaystyle X} {\displaystyle X} and X ∗ {\displaystyle X^{*}} {\displaystyle X^{*}} are endowed with strong topology and weak* topology respectively.
  • There exists r > 0 {\displaystyle r>0} {\displaystyle r>0} such that one can find certain ‖ x ′ ‖ > r {\displaystyle \|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)} {\displaystyle \max \,(\Phi (0),\Phi (x'))<\inf \limits _{\|x\|=r}\Phi (x)=:m(r)}.
  • Φ {\displaystyle \Phi } {\displaystyle \Phi } satisfies weak Palais–Smale condition on { x ∈ X ∣ m ( r ) ≤ Φ ( x ) } {\displaystyle \{x\in X\mid m(r)\leq \Phi (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} {\displaystyle {\overline {x}}\in X} of Φ {\displaystyle \Phi } {\displaystyle \Phi } satisfying m ( r ) ≤ Φ ( x ¯ ) {\displaystyle m(r)\leq \Phi ({\overline {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'\}} {\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)).} {\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

  1. Ambrosetti, Antonio; Rabinowitz, Paul H. (1973). "Dual variational methods in critical point theory and applications". Journal of Functional Analysis. 14 (4): 349–381. doi:10.1016/0022-1236(73)90051-7.
  2. Rabinowitz, Paul H. (1982). "The Mountain Pass Theorem: Theme and Variations". In de Figueiredo, D. G.; Hönig, C. S. (eds.). Differential Equations: Proceedings of the 1st Latin American School of Differential Equations. Lecture Notes in Mathematics. Vol. 957. New York: Springer. pp. 237–271. ISBN 0-387-11951-5.

Further reading