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

Abstract

Let $\Omega ⋐{\mathbb{R}}^n$, $f\in C^1({\mathbb{R}}^{N\times n})$ and $g\in C^1({\mathbb{R}}^N)$, where $N,n\in \mathbb{N}$. We study the minimisation problem of finding $u\in W_0^{1,\infty }(\Omega ;{\mathbb{R}}^N)$ that satisfies \[ \big\| f(\mathrm D u) \big\|_{L^\infty(Ω)} \! = \inf \Big\{\big\| f(\mathrm D v) \big\|_{L^\infty(Ω)} \! : \ v \! \in W^{1,\infty}_0(Ω;\mathbb R^N), \, \| g(v) \|_{L^\infty(Ω)}\! =1\Big\}, \] 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.

30 pages, Journal: Nonlinear Analysis (in press)