Abstract/Summary

This thesis is a collection of published, submitted and developing papers. Each paper is presented as a chapter of this thesis, in each paper we advance the field of vectorial Calculus of Variations in L ∞. This new progress includes constrained problems, such as the constraint of the Navier-Stokes equations studied in Chapter 2. Additionally the combination of constraints, including a nonlinear operator and a supremal functional, deliberated in Chapter 3. Finally, Chapter 4 presents an alternative supremal constraint, in the contemplation of the second order generalised ∞-eigenvalue problem. In Chapter 2 we introduce the joint paper with Nikos Katzourakis and Boris Muha. We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler-Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence form counterpart of the corresponding nondivergence Aronsson-Euler system. In Chapter 3 we present the joint paper with Nikos Katzourakis. We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞, we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. Chapter 4 provides part of the corresponding developing preprint, joint work with Nikos Katzourakis. We consider the problem of minimising the L ∞ norm of a function of the Hessian over a class of maps, subject to a mass constraint involving the L ∞ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of L p approximations, we establish the existence of a special L ∞ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem.