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Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$

Nikos Katzourakis, Roger Moser

Abstract

We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(Ω)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $Ω\Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $\mathrm F_\infty$. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial Ω$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$.

Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$

Abstract

We study variational problems for second order supremal functionals , where satisfies certain natural assumptions, is a positive matrix, and . Higher order problems are very novel in the Calculus of Variations in , and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for . We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for ; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on , and also their partial regularity outside a singular set of codimension one, which may be non-empty even if .
Paper Structure (15 sections, 15 theorems, 229 equations)

This paper contains 15 sections, 15 theorems, 229 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and auxiliary results
  3. Preliminaries and notational conventions
  4. Discussion of assumptions and examples
  5. The restricted convexity assumptions \ref{['1.13']}-\ref{['1.15']}
  6. The Whitney differentiability assumption \ref{['1.17A']}
  7. Explicit classes of supremands
  8. The functional space setup and duality
  9. Completeness and a priori estimates
  10. Identification of the predual of $\mathcal{W}^{2,\infty}_0(\Omega)$ and weak* approximation
  11. Compact sets in $\mathcal{W}^{2,\infty}_0(\Omega)$ and examples
  12. $\mathrm L^p$-limits of $\mathrm L^{\infty}$ constrained minimisers
  13. A lemma of strong compactness in $\mathrm L^1$
  14. Nodal sets and non-trivial solutions
  15. Proofs of the main results

Key Result

Theorem 2

Consider the functional 1.1, where the supremand $F$ is assumed to satisfy 1.6-1.7. Let $u_\infty \in \mathcal{W}^{2,\infty}(\Omega)$ and $f_\infty \in \mathrm L^1(\Omega)\setminus\{0\}$ be given functions which satisfy the system of equations 1.8-1.9. Then, for any compact set $\mathcal{K} \subsete

Theorems & Definitions (20)

  • Definition 1: Global, narrow, local minimisers
  • Theorem 2: Sufficiency of the equations
  • Theorem 3: Sufficiency of the equations and strictness of miminisers
  • Theorem 4: Necessity of the equations
  • Theorem 5: Uniqueness of narrow and global minimisers
  • Corollary 6: Partial regularity
  • Theorem 7: Necessity of the equations and $\mathrm L^p$ approximation with penalisation
  • Corollary 8: Extension to non-differentiable supremands with arbitrary growth
  • Lemma 9
  • Remark 10
  • ...and 10 more