Title:

Some problems in vectorial calculus of variations in L∞

This thesis is a collection of published and submitted papers. Each paper is the chapter of the thesis and new approach involves proving a pointwise characterisation of the vectorial infinity Laplacian in the Chapter 2, proving a geometric feature of the p – Harmonic and ∞ – Harmonic maps in the Chapter 3, finding an explicit ∞ – Harmonic functions in the Chapter 4, proving two distinct minimality principles for a general supremal first order functionals in the Chapter 5. In Chapter 2 we introduce the joint paper with N.Katzourakis, which extends the result of [56]. Let n, N ∈ N with Ω ⊆ R n open. Given H ∈ C 2 (Ω×R N ×R Nn), we consider the functional E∞(u, O) := ess sup O H(·, u, Du), u ∈ W 1,∞ loc (Ω, R N ), O b Ω. (1) The associated PDE system which plays the role of EulerLagrange equations in L ∞ is HP (·, u, Du) D H(·, u, Du) � = 0, H(·, u, Du) [[HP (·, u, Du)]]⊥ � Div HP (·, u, Du) � − Hη(·, u, Du) � = 0, (2) where [[A]]⊥ := ProjR(A)⊥ denotes the orthogonal projection onto the orthogonal complement of the range R(A) ⊆ R N of a linear map A : R n −→ R N . Herein we establish that generalised solutions to (2) can be characterised as local minimisers of (1) for appropriate classes of affine variations of the energy. Generalised solutions to (2) are understood as Dsolutions, a general framework recently introduced by N.Katzourakis in [57, 58]. In Chapter 3 we present the joint paper with N.Katzourakis and H.Abugirda in which we consider 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 pLaplace 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 either the rank of Du is equal to one or n = 2 and u has additively separated form. As a consequence we obtain corresponding flatness results for pHarmonic maps for p ∈ [2,∞]. The aim of the Chapter 4 is to derive new explicit solutions to the ∞Laplace equation, the fundamental PDE arising in Calculus of Variations in the space L ∞. These solutions obey certain symmetry conditions and are derived in arbitrary dimensions, containing as particular subcases the already known classes of twodimensional infinityharmonic functions. Chapter 5 is the joint paper with N.Katzourakis. We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson’s standard notion of absolute minimisers and the concept of ∞minimal maps introduced more recently by N.Katzourakis. We prove that C 1 absolute minimisers characterise a divergence system with parameters probability measures and that C 2 ∞minimal maps characterise Aronsson’s PDE system. Since in the scalar case these different variational concepts coincide, it follows that the nondivergence Aronsson’s equation has an equivalent divergence counterpart.
