Sobolev Spaces with a Vanishing Trace Condition

In this chapter, we define (fractional) Sobolev spaces that vanish on a lower dimensional subset of \(\mathbb {R}^d\) . These lower dimensional sets are so-called \((d-1)\) -regular sets (see Definition 8.1) and include the Ahlfors–David regular sets that we already studied in the last chapter. Eventually, we will also define (fractional) Sobolev spaces on an open set \(O \subseteq \mathbb {R}^d\) with and without a vanishing trace condition. To develop the full theory of these spaces, we introduce the interior thickness condition in Definition 8.15 that plays a fundamental role in this monograph.