The Deformation Theorem

David G. Costa
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David G. Costa: University of Nevada, Las Vegas, Department of Mathematical Sciences

Chapter 3 in An Invitation to Variational Methods in Differential Equations, 2007, pp 19-27 from Springer

Abstract: Abstract Let ϕ: X → ℝ be a C 1 functional on a Banach space X. A number c ∈ ℝ is called a critical value of ϕ if ϕ(u) = c for some critical point u ∈ X. The set of all critical points at the level c is denoted by Kc: $$ K_c = \left\{ {u \in X|\phi ^\prime \left( u \right) = 0, \phi \left( u \right) = c} \right\}. $$ Also, we shall denote by ϕ;c the set of all points in X at levels ≤ c, that is, $$ \phi ^c = \left\{ {u \in X|\phi \left( u \right) \leqslant c} \right\}. $$

Date: 2007
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DOI: 10.1007/978-0-8176-4536-6_3

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