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Extending functions from boundary values plays an important role in variousapplications. In this thesis we consider discrete and continuous formulationsof the problem based on $p$-Laplacians, in particular for $p=\infty$ and tightLipschitz extensions. The thesis gives an overview of the existing theory andprovides some novel results on the approximation of tight Lipschitz extensionsfor vector-valued functions.

Infinity-Laplacians on Scalar- and Vector-Valued Functions and Optimal Lipschitz Extensions on Graphs