# 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. 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.

Conteúdo 1 Declaração 2 Visualization 3 Weaker formulation 4 Referências 5 Further reading Statement The assumptions of the theorem are: {estilo de exibição I} is a functional from a Hilbert space H to the reals, {displaystyle Iin C^{1}(H,mathbb {R} )} e {displaystyle I'} is Lipschitz continuous on bounded subsets of H, {estilo de exibição I} satisfies the Palais–Smale compactness condition, {estilo de exibição I=0} , there exist positive constants r and a such that {estilo de exibição I[você]geq a} E se {displaystyle Vert uVert =r} , and there exists {displaystyle vin H} com {displaystyle Vert vVert >r} de tal modo que {estilo de exibição I[v]leq 0} .

If we define: {displaystyle Gamma ={mathbf {g} em C([0,1];H),vert ,mathbf {g} (0)=0,mathbf {g} (1)=v}} e: {displaystyle c=inf _{mathbf {g} in Gamma }máximo _{0leq tleq 1}EU[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 {estilo de exibição I=0} , and a far-off spot v where {estilo de exibição I[v]leq 0} . In between the two lies a range of mountains (no {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 {estilo de exibição 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 {estilo de exibição X} e {estilo de exibição X^{*}} are endowed with strong topology and weak* topology respectively. There exists {displaystyle r>0} such that one can find certain {estilo de exibição |x'|>r} com {displaystyle max ,(Phi (0),Phi (x'))

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