Sobolev $L^2_p$-functions on closed subsets of $R^2$

TL;DR

This paper characterizes the restrictions of Sobolev functions on finite subsets of R^2 using intrinsic geometric and oscillation-based criteria related to Menger curvature, for p>2.

Contribution

It provides a new intrinsic trace criterion for Sobolev spaces on finite sets in R^2 based on weighted oscillations and Menger curvature, extending previous understanding.

Findings

Intrinsic trace characterization for Sobolev functions on finite sets.

Use of weighted oscillations related to Menger curvature.

Applicable for all p>2.

Abstract

For each $p>2$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^1_p(R^2)$ to an arbitrary finite subset $E$ of $R^2$. The trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the Menger curvature of triangles with vertices in $E$.