On Critical Points of Functionals with Polyconvex Integrands
نویسنده
چکیده
|f(x,A)| ≤ c(1 + |A|), for some c > 0 and 1 ≤ p < ∞, and u lies in the Sobolev space of vector-valued functions W 1,p(Ω,Rm). We study the implications of a function u0 being a critical point of F . In this regard we show among other things that if f does not depend on the spatial variable x, then every piecewise affine critical point of F is a global minimizer subject to its own boundary condition. Moreover for the general case, we construct an example exhibiting that the uniform positivity of the second variation at a critical point is not sufficient for it to be a strong local minimizer. In this example f is discontinuous in x but smooth in A.
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