Vector‐valued optimal Lipschitz extensions

Abstract

Abstract.Consider a bounded open set U ⊂ R n and a Lipschitz function g : ∂U → R m . Does this function always have a canonical optimal Lipschitz extension to all of U ? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.

Minimal Lipschitz and ∞-harmonic extensions of vector-valued functions on finite graphs

“…The existence of L-lex minimal extensions in the non-weighted case was shown in [38,Theorem 1.2]. The existence in the weighted setting as well as for lex minimal extensions can be proved similarly; see [19].…”

“…was considered by [38]. This functional is not strictly convex for p ∈ (1, ∞), but has nevertheless a unique minimizer f ∞,p ∈ H g (V ) of E ∞,p ; see [19].…”

“…The first notation can be found for instance in [26]. The L-lex minimal extension was called tight extension in the paper [38].…”

“…In [38,Section 2] an algorithm for computing these minimal Lipschitz extensions in the scalar-valued case was presented. To the best of our knowledge there exists no meaningful algorithm for computing minimal Lipschitz extensions of vector-valued functions and it is our goal to address their approximation in the present paper.…”

On the numerical approximation of $$\infty $$ ∞ -harmonic mappings

“…We conclude this introduction by noting that in the paper [37] the authors derived a different more singular multi-valued version of "∞-Laplacian" which describes optimal Lipschitz extensions. In our setting this amounts to changing in (1.3) from the Euclidean norm "| · |" we are using on R Nn to the nonsmooth operator norm A = max |a|=1 |Aa|.…”