4 research outputs found
Direct approach to the problem of strong local minima in Calculus of Variations
The paper introduces a general strategy for identifying strong local
minimizers of variational functionals. It is based on the idea that any
variation of the integral functional can be evaluated directly in terms of the
appropriate parameterized measures. We demonstrate our approach on a problem of
W^{1,infinity} weak-* local minima--a slight weakening of the classical notion
of strong local minima. We obtain the first quasiconvexity-based set of
sufficient conditions for W^{1,infinity} weak-* local minima.Comment: 26 pages, no figure
Non-uniqueness of minimizers for strictly polyconvex functionals
In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557–611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals,
where Ω is homeomorphic to a ball.
We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surfaces