Generalised vectorial [formula omitted]-eigenvalue nonlinear problems for [formula omitted] functionals

Abstract

Let Ω⋐Rn, f∈C1(RN×n) and g∈C1(RN), where N,n∈N. We study the minimisation problem of finding u∈W01,∞(Ω;RN) that satisfies ‖f(Du)‖L∞(Ω)=inf{‖f(Dv)‖L∞(Ω):v∈W01,∞(Ω;RN),‖g(v)‖L∞(Ω)=1},under natural assumptions on f,g. This includes the ∞-eigenvalue problem as a special case. Herein we prove the existence of a minimiser u∞ with extra properties, derived as the limit of minimisers of approximating constrained Lp problems as p→∞. A central contribution and novelty of this work is that u∞ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence ∞-Laplacian. Our results are new even in the scalar case of the ∞-eigenvalue problem.