On a vector-valued generalisation of viscosity solutions for general PDE systems

We propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall-Ishii-Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows "nonlinear passage of derivatives" to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this first part of our two-part work we introduce and study the rudiments of this theory, leaving applications for the second part.Comment: 34 pages, 6 figure

On the structure of ∞\infty-Harmonic maps

Let $H \in C^2(\mathbb{R}^{N \times n})$, $H\geq 0$. The PDE system \label{1} A_\infty u \, :=\, \Big(H_P \otimes H_P + H [H_P]^\bot H_{PP} \Big)(Du) : D^2 u\, = \, 0 \tag{1} arises as the ``Euler-Lagrange PDE" of vectorial variational problems for the functional $E_{\infty}(u,\Omega) = \| H(Du) \|_{L^\infty(\Omega)}$ defined on maps $u : \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$. \eqref{1} first appeared in the author's recent work \cite{K3}. The scalar case though has a long history initiated by Aronsson in \cite{A1}. Herein we study the solutions of \eqref{1} with emphasis on the case of $n=2\leq N$ with $H$ the Euclidean norm on $\mathbb{R}^{N \times n}$, which we call the ``$\infty$-Laplacian". By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to $N \geq 2$ the Aronsson-Evans-Yu theorem regarding non-existence of zeros of $|Du|$ and prove a Maximum Principle. We further characterise all $H$ for which \eqref{1} is elliptic and also study the initial value problem for the ODE system arising for $n=1$ but with $H(\cdot,u,u')$ depending on all the arguments.Comment: 30 pages, 10 figures, revised including referees' comments, (Communications in PDE