posted on 2023-06-07, 15:49authored byMohammad Sadegh Shahrokhi-Dehkordi
Let O ? Rn be a bounded Lipschitz domain and consider the energy functional F[u, O] := ? O F(?u(x)) dx, over the space Ap(O) := {u ? W 1,p(O, Rn): u|?O = x, det ?u> 0 a.e. in O}, where the integrand F : Mn×n ? R is quasiconvex, sufficiently regular and satisfies a p-coercivity and p-growth for some exponent p ? [1, 8[. A motivation for the study of above energy functional comes from nonlinear elasticity where F represents the elastic energy of a homogeneous hyperelastic material and Ap(O) represents the space of orientation preserving deformations of O fixing the boundary pointwise. The aim of this thesis is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of F and the relation it bares to the domain topology. Our work, building upon previous works of others, explicitly and quantitatively confirms the significant role of domain topology, and provides explicit and new examples as well as methods for constructing such maps. Our approach for constructing strong local minimizers is topological in nature and is based on defining suitable homotopy classes in Ap(O) (for p = n), whereby minimizing F on each class results in, modulo technicalities, a strong local minimizer. Here we work on a prototypical example of a topologically non-trivial domain, namely, a generalised annulus, O= {x ? Rn : a< |x|