Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems

Nikos Katzourakis

doi:10.3934/cpaa.2015.14.313

Article Contents

2015, Volume 14, Issue 1: 313-327. Doi: 10.3934/cpaa.2015.14.313

This issue Previous Article Global gradient estimates in elliptic problems under minimal data and domain regularity Next Article First eigenfunctions of the 1-Laplacian are viscosity solutions

Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems

Nikos Katzourakis^1,

1.
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX

Received: March 2014

Revised: April 2014

Published: January 2015

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Abstract

For a Hamiltonian $H \in C^2(R^{N \times n})$ and a map $u:\Omega \subseteq R^n \to R^N$, we consider the supremal functional \begin{eqnarray} E_\infty (u,\Omega) := \|H(Du)\|_{L^\infty(\Omega)} . \end{eqnarray} The ``Euler-Lagrange" PDE associated to (1) is the quasilinear system \begin{eqnarray} A_\infty u := (H_P \otimes H_P + H[H_P]^\bot H_{PP})(Du):D^2 u = 0. \end{eqnarray} (1) and (2) are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [28]. Herein we show that the Dirichlet problem for (2) admits for all $n = N \geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [24] has no counterpart when $N \geq 2$. The key idea in the proofs is to recast (2) as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq R^{n\times n}$, $x\in \Omega$.

Keywords:

Nonuniqueness,
vector-valued calculus of variations in $L^\infty$,
$\infty$-Laplacian,
Aronsson PDE,
quasiconformal maps.

Mathematics Subject Classification: Primary: 30C70, 30C75; Secondary: 35J47.

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References

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