In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier "quasi" to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.
- Pub Date:
- November 2012
- Mathematics - Functional Analysis;
- Mathematics - Analysis of PDEs;
- Mathematics - Classical Analysis and ODEs;
- 33 pages. v3: Correction to Example 2.4.3. Specifically, alpha-H\"older continuous functions, for alpha strictly less than one, do not satisfy (P3). Thus one cannot conclude that quasi-AMLEs exist in this case. Please note that the error remains in the published version of the paper in Mathematische Annalen. v2: Several minor corrections and edits, a new appendix (Appendix A)