Sobolev sheaves on the plane

TL;DR

This paper constructs Sobolev sheaves on the plane that match classical Sobolev spaces on cuspidal domains and computes their cohomology, advancing the understanding of sheafification of Sobolev spaces.

Contribution

It introduces Sobolev sheaves on the plane for any integer k and computes their cohomology, linking Sobolev spaces with sheaf theory on definable sites.

Findings

Existence of Sobolev sheaves on the plane for all integer k.

Complete cohomology computation using 'Good direction' concept.

Framework for sheafification of Sobolev spaces in higher dimensions.

Abstract

In this paper, we show that for any integer $k \in \mathbb{N}$ there exists a Sobolev sheaf (in the sense of Lebeau) on any definable site of $\mathbb{R}^2$ that agrees with Sobolev spaces on cuspidal domains. We also provide a complete computation of the cohomology of these sheaves using the notion of 'Good direction' introduced by Valette. This paper serves as an introduction to a more general project on the sheafification of Sobolev spaces in higher dimensions.