A New Characterisation of $\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^{\infty }$

Abstract

Let $u:\Omega \subseteq {\mathbb{R}}^n⟶{\mathbb{R}}^N$ be a smooth map and $n,N\in \mathbb{N}$. The $\infty$-Laplacian is the PDE system \[ \tag{1} \label{1} Δ_\infty u \, :=\, \Big(Du \otimes Du + |Du|^2[Du]^\bot\! \otimes I\Big) :D^2u\, =\, 0, \] where $[Du{]}^{\perp }:={\text{Proj}}_{R(Du{)}^{\perp }}$. \eqref{1} constitutes the fundamental equation of vectorial Calculus of Variations in $L^{\infty }$, associated to the model functional \[ \tag{2} \label{2} E_\infty (u,Ω')\, =\, \big\| |Du|^2\big\|_{L^\infty(Ω')} ,\ \ \ Ω' \Subset Ω. \] We show that generalised solutions to \eqref{1} can be characterised in terms of \eqref{2} via a set of designated affine variations. For the scalar case $N=1$, we utilise the theory of viscosity solutions of Crandall-Ishii-Lions. For the vectorial case $N\ge 2$, we utilise the recently proposed by the author theory of $\mathcal{D}$-solutions. Moreover, we extend the result described above to the $p$-Laplacian, $1<p<\infty$.

20 pages; El. Journal of Differential Equations, 2017

References in corpus (5)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems
• $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^{\infty }$ via the singular value problem
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Cited by in corpus (7)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian
• A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^{\infty }$
• On the Numerical Approximation of $\infty$-Harmonic Mappings
• Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^{\infty }$
• A Remark on Global $W^{1,p}$ Bounds for Harmonic Functions with Lipschitz Boundary Values
• Explicit $\infty$-harmonic functions in high dimensions