Deformation of functionals having a unique critical point
type: Item
instance of: scholarly article
title: Deformation of functionals having a unique critical point
zbMATH Open document ID: 0546.49002
DOI: 10.1007/BF01140348
author: N. A. Bobylev
published in: Mathematical Notes
publication date: 1983-01-01T00:00:00Z
review text: For a class of functionals defined on a Hilbert space the following theorem is proved: Let us assume that the functionals \(f_ 0\) and \(f_ 1\) can be connected by means of a nondegenerate deformation. Suppose that \(x=0\) is a point of local minimum of the functional \(f_ 0\). Then \(x=0\) is also a local minimum of the functional \(f_ 1\). The nondegenerate deformation is a continuous functional f(x,\(\lambda)\), \(x\in T=\{x\in H: \| x\|\leq 1\}\), 0\(\leq\lambda \leq 1\), such that \(f_ 0(x)=f(x,0)\), \(f_ 1(x)=f(x,1)\), satisfying some additional regularity conditions. Interesting applications of the above theorem are suggested. These applications pertain to the fundamental classical inequalities and to integral functionals of the calculus of variations. For instance, a stronger version of Wirtinger's inequality is derived.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 3870072
zbMATH Keywords: nondegenerate deformation
zbMATH Keywords: integral functionals
zbMATH Keywords: Wirtinger's inequality
MaRDI profile type: Publication
cites work: Nonlinear eigenvalue problems and Galerkin approximations
cites work: Nonlinear Eigenvalue Problems and Group Invariance
cites work: Q4049449
cites work: Uniqueness of Solutions of Boundary Value Problems
full work available at URL: https://doi.org/10.1007/bf01140348
OpenAlex ID: W2318453913
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description: scientific article; zbMATH DE number 3870072