Abstract/Summary

Let Ω R n , f ∈ C 1 (R N ×n) and g ∈ C 1 (R N), where N, n ∈ N. We study the minimisation problem of finding u ∈ W 1,∞ 0 (Ω; R N) that satisfies f (Du) L ∞ (Ω) = inf f (Dv) L ∞ (Ω) : v ∈ W 1,∞ 0 (Ω; R N), 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 L p 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.