Generalised vectorial $\infty$-eigenvalue nonlinear problems for $L^\infty$ functionals

TL;DR

This paper studies a generalized vectorial $ abla^ ext{infty}$-eigenvalue problem, establishing existence of solutions as limits of $L^p$ minimizers, and introduces a divergence PDE with measure coefficients extending the scalar $ ext{infty}$-Laplacian.

Contribution

It extends the $ ext{infty}$-eigenvalue problem to vector-valued functions, proving existence of solutions via limits of $L^p$ problems and deriving a novel divergence PDE with measure coefficients.

Findings

Existence of minimizers as $p o \infty$

Derivation of a divergence PDE with measure coefficients

Extension of scalar $ ext{infty}$-eigenvalue results to vector case

Abstract

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\| f(\mathrm D u) \big\|_{L^\infty(\Omega)} \! = \inf \Big\{\big\| f(\mathrm D v) \big\|_{L^\infty(\Omega)} \! : \ v \! \in W^{1,\infty}_0(\Omega;\mathbb R^N), \, \| g(v) \|_{L^\infty(\Omega)}\! =1\Big\}, \] under natural assumptions on $f,g$. This includes the $\infty$-eigenvalue problem as a special case. Herein we prove existence of a minimiser $u_\infty$ with extra properties, derived as the limit of minimisers of approximating constrained $L^p$ problems as $p\to \infty$. A central contribution and novelty of this work is that $u_\infty$ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence $\infty$-Laplacian. Our results are new even in the scalar case of the $\infty$-eigenvalue problem.