Regularity and uniqueness in the calculus of variations

Thesis

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∈W^1,p_g(Ω,R^N) for some boundary datum g∈C^1,α(‾Ω, R^N).

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∇gΙΙ_LP < 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∇gΙΙ_O,α < M. Furthermore, when the domain Ω is star-shaped, we extend the regularity result to the case of W^1,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 W^1,p_g 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.

Authors: Campos Cordero, J

• Publication date: 2014
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: integral functionals; minimizers
• Subjects: Mathematics