Mountain pass theorem

Mountain pass theorem The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz.[1] 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.

Contents 1 Statement 2 Visualization 3 Weaker formulation 4 References 5 Further reading Statement The assumptions of the theorem are: {displaystyle I} is a functional from a Hilbert space H to the reals, {displaystyle Iin C^{1}(H,mathbb {R} )} and {displaystyle I'} is Lipschitz continuous on bounded subsets of H, {displaystyle I} satisfies the Palais–Smale compactness condition, {displaystyle I[0]=0} , there exist positive constants r and a such that {displaystyle I[u]geq a} if {displaystyle Vert uVert =r} , and there exists {displaystyle vin H} with {displaystyle Vert vVert >r} such that {displaystyle I[v]leq 0} .

If we define: {displaystyle Gamma ={mathbf {g} in C([0,1];H),vert ,mathbf {g} (0)=0,mathbf {g} (1)=v}} and: {displaystyle c=inf _{mathbf {g} in Gamma }max _{0leq tleq 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 {displaystyle I[0]=0} , and a far-off spot v where {displaystyle I[v]leq 0} . In between the two lies a range of mountains (at {displaystyle Vert uVert =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 {displaystyle X} be Banach space. The assumptions of the theorem are: {displaystyle Phi in C(X,mathbf {R} )} and have a Gateaux derivative {displaystyle Phi 'colon Xto X^{*}} which is continuous when {displaystyle X} and {displaystyle X^{*}} are endowed with strong topology and weak* topology respectively. There exists {displaystyle r>0} such that one can find certain {displaystyle |x'|>r} with {displaystyle max ,(Phi (0),Phi (x'))

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