CPAA
Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems
Nikos Katzourakis
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 Aronsson PDE $\infty$-Laplacian vector-valued calculus of variations in $L^\infty$ quasiconformal maps.
DCDS
$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
Gisella Croce Nikos Katzourakis Giovanni Pisante
For
$\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$
and
$u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$
, consider the system
$ \label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,0. \tag{1}$
We construct
$\mathcal{D}$
-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our
$\mathcal{D}$
-solutions are
$W^{1,∞}$
-submersions and are obtained without any convexity hypotheses for
$\mathrm{H}$
, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions
$n≠ N$
.
keywords: Vectorial Calculus of Variations in L generalised solutions fully nonlinear systems ∞-Laplacian young measures singular value problem Baire Category method convex integration

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