Abstract/Summary

This thesis is a collection of published and submitted papers. Each paper presents a chapter of the thesis and in each paper we make progress in the field of nondivergence systems of nonlinear PDEs. The new progress includes proving the existence and uniqueness of strong solutions to first order elliptic systems in Chapter 2, proving the existence of absolute minimisers to a vectorial 1D minimisation problem in Chapter 3, proving geometric aspects of p-Harmonic maps in Chapter 4, proving new properties of classical solutions to the vectorial infinity Laplacian in Chapter 5. In Chapter 2 of this thesis we present the joint paper with Katzourakis in which we extend the results of [43]. In the very recent paper [43], Katzourakis proved that for any f ∈ L 2 (R n , R N ), the fully nonlinear first order system F(·, Du) = f is well posed in the so-called J.L. Lions space and moreover the unique strong solution u : R n −→ R N to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for F inspired by Campanato’s classical work in the 2nd order case. Herein we extend the results of [43] by introducing a new strictly weaker ellipticity condition and by proving well posedness in the same “energy” space. In Chapter 3 of this thesis we present the joint paper with Katzourakis in which we prove the existence of vectorial Absolute Minimisers in the sense of Aronsson for the supremal functional E∞(u, Ω 0 ) = kL (·, u, Du)kL∞(Ω0) , Ω0 b Ω, applied to W1,∞ maps u : Ω ⊆ R −→ R N with given boundary values. The assumptions on L are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by Katzourakis. In Chapter 4 of this thesis we present the joint paper with Katzourakis and Ayanbayev in which we consider the PDE system of vanishing normal projection of the Laplacian for C 2 maps u : R n ⊇ Ω −→ R N : [[Du]]⊥∆u = 0 in Ω. This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the p-Laplace system for all p ∈ [2,∞]. For p = ∞, the ∞-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in L ∞. Herein we show that the image of a solution u is piecewise affine if the rank of Du is equal to one. As a consequence we obtain corresponding flatness results for p-Harmonic maps, p ∈ [2,∞]. In Chapter 5 of this thesis we present a single authored paper in which we discuss an extension of a recent paper [41]. In [41], among other interesting results, Katzourakis analysed the phenomenon of separation of the solutions u:R 2 ⊇ Ω −→ R N , to the ∞-Laplace system ∆∞u := ( Du ⊗ Du + |Du| 2 [[Du]]⊥ ⊗ I ) : D2u = 0, to phases with qualitatively different behaviour in the case of n = 2 ≤ N. The solutions of the ∞-Laplace system are called the ∞-Harmonic mappings. Chapter 5 of this thesis present an extension of Katzourakis’ result mentioned above to higher dimensions by studying the phase separation of n-dimensional ∞-Harmonic mappings in the case N ≥ n ≥ 2.