Deformation of functionals having a unique critical point
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.
scientific article; zbMATH DE number 3870072