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Title: Regularity and uniqueness in the calculus of variations
Author: Campos Cordero, Judith
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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This thesis is about regularity and uniqueness of minimizers of integral functionals of the form F(u) := ∫Ω F(∇u(x)) dx; where F∈C2(RNn) is a strongly quasiconvex integrand with p-growth, Ω⊆RnRn is an open bounded domain and u∈W1,pg(Ω,RN) for some boundary datum g∈C1,α(‾Ω, RN). The first contribution of this work is a full regularity result, up to the boundary, for global minimizers of F provided that the boundary condition g satisfies that ΙΙ∇gΙΙLP < ε for some ε > 0 depending only on n;N, the parameters given by the strong quasiconvexity and p-growth conditions and, most importantly, on an arbitrary but fixed constant M > 0 for which we require that ΙΙ∇gΙΙO,α < M. Furthermore, when the domain Ω is star-shaped, we extend the regularity result to the case of W1,p-local minimizers. On the other hand, for the case of global minimizers we exploit the compactness provided by the aforementioned regularity result to establish the main contribution of this thesis: we prove that, under essentially the same smallness assumptions over the boundary condition g that we mentioned above, the minimizer of F in W1,pg is unique. This result appears in contrast to the non-uniqueness examples previously given by Spadaro [Spa09], for which the boundary conditions are required to be suitably large.
Supervisor: Kristensen, Jan Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; integral functionals ; minimizers